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Modular arithmetic

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the concept that uses the "a (mod m)" notation. For the binary operation mod(a,m), see <a href="Modulo" title="Modulo">Modulo</a>.</div>

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In <a href="Mathematics" title="Mathematics">mathematics</a>, modular arithmetic is a system of <a href="Arithmetic" title="Arithmetic">arithmetic</a> operations for <a href="Integer" title="Integer">integers</a>, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> in his book <a href="Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a>, published in 1801.

A familiar example of modular arithmetic is the hour hand on a <a href="12-hour_clock" title="12-hour clock">12-hour clock</a>. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in 7 + 8 = 15, but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is congruent to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12).

Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This can be written as 2 × 8 ≡ 4 (mod 12). Note that after a wait of exactly 12 hours, the hour hand will always be right where it was before, so 12 acts the same as zero, thus 12 ≡ 0 (mod 12).

<div class="mw-heading mw-heading2"><h2 id="Congruence">Congruence</h2></div>

Given an <a href="Integer" title="Integer">integer</a> m ≥ 1, called a modulus, two integers a and b are said to be congruent modulo m, if m is a <a href="Divisor" title="Divisor">divisor</a> of their difference; that is, if there is an integer k such that

Congruence modulo m is a <a href="Congruence_relation" title="Congruence relation">congruence relation</a>, meaning that it is an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> that is compatible with <a href="Addition" title="Addition">addition</a>, <a href="Subtraction" title="Subtraction">subtraction</a>, and <a href="Multiplication" title="Multiplication">multiplication</a>. Congruence modulo m is denoted by

The parentheses mean that (mod m) applies to the entire equation, not just to the right-hand side (here, b).

This notation is not to be confused with the notation b mod m (without parentheses), which refers to the remainder of b when divided by m, known as the <a href="Modulo" title="Modulo">modulo</a> operation: that is, b mod m denotes the unique integer r such that 0 ≤ r < m and r ≡ b (mod m).

The congruence relation may be rewritten as

explicitly showing its relationship with <a href="Euclidean_division" title="Euclidean division">Euclidean division</a>. However, the b here need not be the remainder in the division of a by m. Rather, a ≡ b (mod m) asserts that a and b have the same <a href="Remainder" title="Remainder">remainder</a> when divided by m. That is,

where 0 ≤ r < m is the common remainder. We recover the previous relation (a − b = k m) by subtracting these two expressions and setting k = p − q.

Because the congruence modulo m is defined by the <a href="Divisor#Further_notions_and_facts" title="Divisor">divisibility</a> by m and because −1 is a <a href="Unit_(ring_theory)#Integer_ring" title="Unit (ring theory)">unit</a> in the ring of integers, a number is divisible by −m exactly if it is divisible by m.

This means that every non-zero integer m may be taken as modulus.

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

In modulus 12, one can assert that:

because the difference is 38 − 14 = 24 = 2 × 12, a multiple of 12. Equivalently, 38 and 14 have the same remainder 2 when divided by 12.

The definition of congruence also applies to negative values. For example:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}2&\equiv -3{\pmod {5}}\\-8&\equiv {\phantom {+}}7{\pmod {5}}\\-3&\equiv -8{\pmod {5}}.\end{aligned}}}</annotation>

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</math><img src="./96427ea4e4951105b655e6af2c127031c733132e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.284ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}2&\equiv -3{\pmod {5}}\\-8&\equiv {\phantom {+}}7{\pmod {5}}\\-3&\equiv -8{\pmod {5}}.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Basic_properties">Basic properties</h2></div>

The congruence relation satisfies all the conditions of an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a>:

<ul><li>Reflexivity: a ≡ a (mod m)</li>

<li>Symmetry: a ≡ b (mod m) if b ≡ a (mod m).</li>

<li>Transitivity: If a ≡ b (mod m) and b ≡ c (mod m), then a ≡ c (mod m)</li></ul>

If a1 ≡ b1 (mod m) and a2 ≡ b2 (mod m), or if a ≡ b (mod m), then:<a href="#cite_note-1">[1]</a>

<ul><li>a + k ≡ b + k (mod m) for any integer k (compatibility with translation)</li>

<li>k a ≡ k b (mod m) for any integer k (compatibility with scaling)</li>

<li>k a ≡ k b (mod k m) for any integer k</li>

<li>a1 + a2 ≡ b1 + b2 (mod m) (compatibility with addition)</li>

<li>a1 − a2 ≡ b1 − b2 (mod m) (compatibility with subtraction)</li>

<li>a1 a2 ≡ b1 b2 (mod m) (compatibility with multiplication)</li>

<li>ak ≡ bk (mod m) for any non-negative integer k (compatibility with exponentiation)</li>

<li>p(a) ≡ p(b) (mod m), for any <a href="Polynomial" title="Polynomial">polynomial</a> p(x) with integer coefficients (compatibility with polynomial evaluation)</li></ul>

If a ≡ b (mod m), then it is generally false that ka ≡ kb (mod m). However, the following is true:

<ul><li>If c ≡ d (mod φ(m)), where φ is <a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a>, then ac ≡ ad (mod m)—provided that a is <a href="Coprime" class="mw-redirect" title="Coprime">coprime</a> with m.</li></ul>

For cancellation of common terms, we have the following rules:

<ul><li>If a + k ≡ b + k (mod m), where k is any integer, then a ≡ b (mod m).</li>

<li>If k a ≡ k b (mod m) and k is coprime with m, then a ≡ b (mod m).</li>

The last rule can be used to move modular arithmetic into division. If b divides a, then (a/b) mod m = (a mod b m) / b.

The <a href="Modular_multiplicative_inverse" title="Modular multiplicative inverse">modular multiplicative inverse</a> is defined by the following rules:

<ul><li>Existence: There exists an integer denoted a−1 such that aa−1 ≡ 1 (mod m) if and only if a is coprime with m. This integer a−1 is called a modular multiplicative inverse of a modulo m.</li>

<li>If a ≡ b (mod m) and a−1 exists, then a−1 ≡ b−1 (mod m) (compatibility with multiplicative inverse, and, if a = b, uniqueness modulo m).</li>

<li>If ax ≡ b (mod m) and a is coprime to m, then the solution to this linear congruence is given by x ≡ a−1b (mod m).</li></ul>

The multiplicative inverse x ≡ a−1 (mod m) may be efficiently computed by solving <a href="B%C3%A9zout's_identity" title="Bézout's identity">Bézout's equation</a> a x + m y = 1 for x, y, by using the <a href="Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">Extended Euclidean algorithm</a>.

In particular, if p is a prime number, then a is coprime with p for every a such that 0 < a < p; thus a multiplicative inverse exists for all a that is not congruent to zero modulo p.

<div class="mw-heading mw-heading2"><h2 id="Advanced_properties">Advanced properties</h2></div>

Some of the more advanced properties of congruence relations are the following:

<ul><li><a href="Fermat's_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a>: If p is prime and does not divide a, then ap−1 ≡ 1 (mod p).</li>

<li><a href="Euler's_theorem" title="Euler's theorem">Euler's theorem</a>: If a and m are coprime, then aφ(m) ≡ 1 (mod m), where φ is <a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a>.</li>

<li>A simple consequence of Fermat's little theorem is that if p is prime, then a−1 ≡ ap−2 (mod p) is the multiplicative inverse of 0 < a < p. More generally, from Euler's theorem, if a and m are coprime, then a−1 ≡ aφ(m)−1 (mod m). Hence, if ax ≡ 1 (mod m), then x ≡ aφ(m)−1 (mod m).</li>

<li>Another simple consequence is that if a ≡ b (mod φ(m)), where φ is Euler's totient function, then ka ≡ kb (mod m) provided k is <a href="Coprime" class="mw-redirect" title="Coprime">coprime</a> with m.</li>

<li><a href="Wilson's_theorem" title="Wilson's theorem">Wilson's theorem</a>: p is prime if and only if (p − 1)! ≡ −1 (mod p).</li>

<li><a href="Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a>: For any a, b and coprime m, n, there exists a unique x (mod mn) such that x ≡ a (mod m) and x ≡ b (mod n). In fact, x ≡ b mn−1 m + a nm−1 n (mod mn) where mn−1 is the inverse of m modulo n and nm−1 is the inverse of n modulo m.</li>

<li><a href="Lagrange's_theorem_(number_theory)" title="Lagrange's theorem (number theory)">Lagrange's theorem</a>: If p is prime and f (x) = a0 xd + ... + ad is a <a href="Polynomial" title="Polynomial">polynomial</a> with integer coefficients such that p is not a divisor of a0, then the congruence f (x) ≡ 0 (mod p) has at most d non-congruent solutions.</li>

<li><a href="Primitive_root_modulo_n" title="Primitive root modulo n">Primitive root modulo m</a>: A number g is a primitive root modulo m if, for every integer a coprime to m, there is an integer k such that gk ≡ a (mod m). A primitive root modulo m exists if and only if m is equal to 2, 4, pk or 2pk, where p is an odd prime number and k is a positive integer. If a primitive root modulo m exists, then there are exactly φ(φ(m)) such primitive roots, where φ is the Euler's totient function.</li>

<li><a href="Quadratic_residue" title="Quadratic residue">Quadratic residue</a>: An integer a is a quadratic residue modulo m, if there exists an integer x such that x2 ≡ a (mod m). <a href="Euler's_criterion" title="Euler's criterion">Euler's criterion</a> asserts that, if p is an odd prime, and a is not a multiple of p, then a is a quadratic residue modulo p if and only if

<div class="mw-heading mw-heading2"><h2 id="Congruence_classes">Congruence classes </h2></div>

The congruence relation is an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a>. The <a href="Equivalence_class" title="Equivalence class">equivalence class</a> modulo m of an integer a is the set of all integers of the form a + k m, where k is any integer. It is called the congruence class or residue class of a modulo m, and may be denoted (a mod m), or as a or [a] when the modulus m is known from the context.

Each residue class modulo m contains exactly one integer in the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,...,|m|-1}">

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</math><img src="./c16b9b50a1528ea7008e29033c02f770ed79678c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.669ex; height:2.843ex;" alt="{\displaystyle 0,...,|m|-1}" loading="lazy">. Thus, these <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |m|}">

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</math><img src="./f9aa0b9c6f0a110ae299bf81e924412842ce2b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.334ex; height:2.843ex;" alt="{\displaystyle |m|}" loading="lazy"> integers are <a href="Representative_(mathematics)" class="mw-redirect" title="Representative (mathematics)">representatives</a> of their respective residue classes.

It is generally easier to work with integers than sets of integers; that is, the representatives most often considered, rather than their residue classes.

Consequently, (a mod m) denotes generally the unique integer r such that 0 ≤ r < m and r ≡ a (mod m); it is called the residue of a modulo m.

In particular, (a mod m) = (b mod m) is equivalent to a ≡ b (mod m), and this explains why "=" is often used instead of "≡" in this context.

<div class="mw-heading mw-heading2"><h2 id="Residue_systems">Residue systems</h2></div>

Each residue class modulo m may be represented by any one of its members, although we usually represent each residue class by the smallest nonnegative integer which belongs to that class<a href="#cite_note-2">[2]</a> (since this is the proper remainder which results from division). Any two members of different residue classes modulo m are incongruent modulo m. Furthermore, every integer belongs to one and only one residue class modulo m.<a href="#cite_note-3">[3]</a>

The set of integers {0, 1, 2, ..., m − 1} is called the least residue system modulo m. Any set of m integers, no two of which are congruent modulo m, is called a complete residue system modulo m.

The least residue system is a complete residue system, and a complete residue system is simply a set containing precisely one <a href="Representative_(mathematics)" class="mw-redirect" title="Representative (mathematics)">representative</a> of each residue class modulo m.<a href="#cite_note-4">[4]</a> For example, the least residue system modulo 4 is {0, 1, 2, 3}. Some other complete residue systems modulo 4 include:

Some sets that are not complete residue systems modulo 4 are:

<ul><li>{−5, 0, 6, 22}, since 6 is congruent to 22 modulo 4.</li>

<li>{5, 15}, since a complete residue system modulo 4 must have exactly 4 incongruent residue classes.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Reduced_residue_systems">Reduced residue systems</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Reduced_residue_system" title="Reduced residue system">Reduced residue system</a></div>

Given the <a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a> φ(m), any set of φ(m) integers that are <a href="Coprime_integers" title="Coprime integers">relatively prime</a> to m and mutually incongruent under modulus m is called a reduced residue system modulo m.<a href="#cite_note-5">[5]</a> The set {5, 15} from above, for example, is an instance of a reduced residue system modulo 4.

<div class="mw-heading mw-heading3"><h3 id="Covering_systems">Covering systems</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Covering_system" title="Covering system">Covering system</a></div>

Covering systems represent yet another type of residue system that may contain residues with varying moduli.

<div class="mw-heading mw-heading2"><h2 id="Integers_modulo_m">Integers modulo m</h2></div>

In the context of this paragraph, the modulus m is almost always taken as positive.

The set of all <a href="#Congruence_classes">congruence classes</a> modulo m is a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> called the ring of integers modulo m, and is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {Z} /m\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\textstyle \mathbb {Z} /m\mathbb {Z} }</annotation>

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</math><img src="./0a08c684b0f13affb387daa9275f0f67512eda7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.304ex; height:2.843ex;" alt="{\textstyle \mathbb {Z} /m\mathbb {Z} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /m}</annotation>

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</math><img src="./8ba3ed3e693048c5108f9daf9164b2580ec21883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.753ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /m}" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{m}}">

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The ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m\mathbb {Z} }">

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(In some parts of <a href="Number_theory" title="Number theory">number theory</a> the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{m}}">

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For m > 0 one has

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /m\mathbb {Z} =\left\{{\overline {a}}_{m}\mid a\in \mathbb {Z} \right\}=\left\{{\overline {0}}_{m},{\overline {1}}_{m},{\overline {2}}_{m},\ldots ,{\overline {m{-}1}}_{m}\right\}.}</annotation>

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</math><img src="./c7952b2b1d39f18594aa833834fa43668b47197f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.442ex; height:4.843ex;" alt="{\displaystyle \mathbb {Z} /m\mathbb {Z} =\left\{{\overline {a}}_{m}\mid a\in \mathbb {Z} \right\}=\left\{{\overline {0}}_{m},{\overline {1}}_{m},{\overline {2}}_{m},\ldots ,{\overline {m{-}1}}_{m}\right\}.}" loading="lazy"></dd></dl>

When m = 1, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m\mathbb {Z} }">

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</math><img src="./6fef120a86ce66f277c2cebb01f9c4c14de784de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.304ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /m\mathbb {Z} }" loading="lazy"> is the <a href="Zero_ring" title="Zero ring">zero ring</a>; when m = 0, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /m\mathbb {Z} }</annotation>

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</math><img src="./6fef120a86ce66f277c2cebb01f9c4c14de784de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.304ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /m\mathbb {Z} }" loading="lazy"> is not an <a href="Empty_set" title="Empty set">empty set</a>; rather, it is <a href="Isomorphism" title="Isomorphism">isomorphic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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Addition, subtraction, and multiplication are defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m\mathbb {Z} }">

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {a}}_{m}+{\overline {b}}_{m}={\overline {(a+b)}}_{m}}</annotation>

</semantics>

</math><img src="./7e1b0ab102f3d880d4e9417633cc1e1facf69dcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.413ex; height:3.843ex;" alt="{\displaystyle {\overline {a}}_{m}+{\overline {b}}_{m}={\overline {(a+b)}}_{m}}" loading="lazy"></li>

<msub>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {a}}_{m}-{\overline {b}}_{m}={\overline {(a-b)}}_{m}}</annotation>

</semantics>

</math><img src="./6e1047bafa007e27ec6252c93f06006b24b7662e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.413ex; height:3.843ex;" alt="{\displaystyle {\overline {a}}_{m}-{\overline {b}}_{m}={\overline {(a-b)}}_{m}}" loading="lazy"></li>

<msub>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {a}}_{m}{\overline {b}}_{m}={\overline {(ab)}}_{m}.}</annotation>

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</math><img src="./2cef0bc14b824350bf2593a5efffda2a0aa43c19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.379ex; height:3.843ex;" alt="{\displaystyle {\overline {a}}_{m}{\overline {b}}_{m}={\overline {(ab)}}_{m}.}" loading="lazy"></li></ul>

The properties given before imply that, with these operations, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m\mathbb {Z} }">

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /m\mathbb {Z} }</annotation>

</semantics>

</math><img src="./6fef120a86ce66f277c2cebb01f9c4c14de784de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.304ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /m\mathbb {Z} }" loading="lazy"> is a <a href="Commutative_ring" title="Commutative ring">commutative ring</a>. For example, in the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /24\mathbb {Z} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /24\mathbb {Z} }</annotation>

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</math><img src="./6b575ee86953e53aa06382cd7289a13fcc2a4583.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.588ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /24\mathbb {Z} }" loading="lazy">, one has

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {12}}_{24}+{\overline {21}}_{24}={\overline {33}}_{24}={\overline {9}}_{24}}</annotation>

</semantics>

</math><img src="./2816d2095bb867f0c38c120070bd87e5793d0f30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.139ex; height:3.176ex;" alt="{\displaystyle {\overline {12}}_{24}+{\overline {21}}_{24}={\overline {33}}_{24}={\overline {9}}_{24}}" loading="lazy"></dd></dl>

as in the arithmetic for the 24-hour clock.

The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m\mathbb {Z} }">

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /m\mathbb {Z} }</annotation>

</semantics>

</math><img src="./6fef120a86ce66f277c2cebb01f9c4c14de784de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.304ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /m\mathbb {Z} }" loading="lazy"> is used because this ring is the <a href="Quotient_ring" title="Quotient ring">quotient ring</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

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</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> by the <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\mathbb {Z} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle m\mathbb {Z} }</annotation>

</semantics>

</math><img src="./cb660446d851fca07d1cfdd36a71c022ad617d23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle m\mathbb {Z} }" loading="lazy">, the set formed by all multiples of m, i.e., all numbers k m with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} .}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} .}</annotation>

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</math><img src="./0829895a7f6a03b7e80aac56c7df6e277fd09cdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.249ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} .}" loading="lazy">

Under addition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m\mathbb {Z} }">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /m\mathbb {Z} }</annotation>

</semantics>

</math><img src="./6fef120a86ce66f277c2cebb01f9c4c14de784de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.304ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /m\mathbb {Z} }" loading="lazy"> is a <a href="Cyclic_group" title="Cyclic group">cyclic group</a>. All finite cyclic groups are isomorphic with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m\mathbb {Z} }">

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /m\mathbb {Z} }</annotation>

</semantics>

</math><img src="./6fef120a86ce66f277c2cebb01f9c4c14de784de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.304ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /m\mathbb {Z} }" loading="lazy"> for some m.<a href="#cite_note-7">[7]</a>

The ring of integers modulo m is a <a href="Field_(mathematics)" title="Field (mathematics)">field</a>, i.e., every nonzero element has a <a href="Modular_multiplicative_inverse" title="Modular multiplicative inverse">multiplicative inverse</a>, if and only if m is <a href="Prime_number" title="Prime number">prime</a>. If m = pk is a <a href="Prime_power" title="Prime power">prime power</a> with k > 1, there exists a unique (up to isomorphism) finite field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GF} (m)=\mathbb {F} _{m}}">

</mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (m)=\mathbb {F} _{m}}</annotation>

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</math><img src="./8717eb7ee202b16ce677e20f835b6ea5370fe792.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.386ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (m)=\mathbb {F} _{m}}" loading="lazy"> with m elements, which is not isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m\mathbb {Z} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /m\mathbb {Z} }</annotation>

</semantics>

If m > 1, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }}">

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }}</annotation>

</semantics>

</math><img src="./eb072a4530d1efb370e5fd8e5c7cc0e03272ed2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.624ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }}" loading="lazy"> denotes the <a href="Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">multiplicative group of the integers modulo m</a> that are invertible. It consists of the congruence classes am, where a <a href="Coprime_integers" title="Coprime integers">is coprime</a> to m; these are precisely the classes possessing a multiplicative inverse. They form an <a href="Abelian_group" title="Abelian group">abelian group</a> under multiplication; its order is φ(m), where φ is <a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a>.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

In pure mathematics, modular arithmetic is one of the foundations of <a href="Number_theory" title="Number theory">number theory</a>, touching on almost every aspect of its study, and it is also used extensively in <a href="Group_theory" title="Group theory">group theory</a>, <a href="Ring_theory" title="Ring theory">ring theory</a>, <a href="Knot_theory" title="Knot theory">knot theory</a>, and <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a>. In applied mathematics, it is used in <a href="Computer_algebra" title="Computer algebra">computer algebra</a>, <a href="Cryptography" title="Cryptography">cryptography</a>, <a href="Computer_science" title="Computer science">computer science</a>, <a href="Chemistry" title="Chemistry">chemistry</a> and the <a href="Visual_arts" title="Visual arts">visual</a> and <a href="Music" title="Music">musical</a> arts.

A very practical application is to calculate checksums within serial number identifiers. For example, <a href="International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">International Standard Book Number</a> (ISBN) uses modulo 11 (for 10-digit ISBN) or modulo 10 (for 13-digit ISBN) arithmetic for error detection. Likewise, <a href="International_Bank_Account_Number" title="International Bank Account Number">International Bank Account Numbers</a> (IBANs) use modulo 97 arithmetic to spot user input errors in bank account numbers. In chemistry, the last digit of the <a href="CAS_registry_number" class="mw-redirect" title="CAS registry number">CAS registry number</a> (a unique identifying number for each chemical compound) is a <a href="Check_digit" title="Check digit">check digit</a>, which is calculated by taking the last digit of the first two parts of the CAS registry number times 1, the previous digit times 2, the previous digit times 3 etc., adding all these up and computing the sum modulo 10.

In cryptography, modular arithmetic directly underpins <a href="Public-key_cryptography" title="Public-key cryptography">public key</a> systems such as <a href="RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA</a> and <a href="Diffie%E2%80%93Hellman_key_exchange" title="Diffie–Hellman key exchange">Diffie–Hellman</a>, and provides <a href="Finite_field" title="Finite field">finite fields</a> which underlie <a href="Elliptic_curve" title="Elliptic curve">elliptic curves</a>, and is used in a variety of <a href="Symmetric_key_algorithm" class="mw-redirect" title="Symmetric key algorithm">symmetric key algorithms</a> including <a href="Advanced_Encryption_Standard" title="Advanced Encryption Standard">Advanced Encryption Standard</a> (AES), <a href="International_Data_Encryption_Algorithm" title="International Data Encryption Algorithm">International Data Encryption Algorithm</a> (IDEA), and <a href="RC4" title="RC4">RC4</a>. RSA and Diffie–Hellman use <a href="Modular_exponentiation" title="Modular exponentiation">modular exponentiation</a>.

In computer algebra, modular arithmetic is commonly used to limit the size of integer coefficients in intermediate calculations and data. It is used in <a href="Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">polynomial factorization</a>, a problem for which all known efficient algorithms use modular arithmetic. It is used by the most efficient implementations of <a href="Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">polynomial greatest common divisor</a>, exact <a href="Linear_algebra" title="Linear algebra">linear algebra</a> and <a href="Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a> algorithms over the integers and the rational numbers. As posted on <a href="Fidonet" class="mw-redirect" title="Fidonet">Fidonet</a> in the 1980s and archived at <a href="Rosetta_Code" title="Rosetta Code">Rosetta Code</a>, modular arithmetic was used to disprove <a href="Euler's_sum_of_powers_conjecture" title="Euler's sum of powers conjecture">Euler's sum of powers conjecture</a> on a <a href="Sinclair_QL" title="Sinclair QL">Sinclair QL</a> <a href="Microcomputer" title="Microcomputer">microcomputer</a> using just one-fourth of the integer precision used by a <a href="CDC_6600" title="CDC 6600">CDC 6600</a> <a href="Supercomputer" title="Supercomputer">supercomputer</a> to disprove it two decades earlier via a <a href="Brute_force_search" class="mw-redirect" title="Brute force search">brute force search</a>.<a href="#cite_note-8">[8]</a>

In computer science, modular arithmetic is often applied in <a href="Bitwise_operation" title="Bitwise operation">bitwise operations</a> and other operations involving fixed-width, cyclic <a href="Data_structure" title="Data structure">data structures</a>. The modulo operation, as implemented in many <a href="Programming_language" title="Programming language">programming languages</a> and <a href="Calculator" title="Calculator">calculators</a>, is an application of modular arithmetic that is often used in this context. The logical operator <a href="XOR" class="mw-redirect" title="XOR">XOR</a> sums 2 bits, modulo 2.

The use of <a href="Long_division" title="Long division">long division</a> to turn a fraction into a <a href="Repeating_decimal" title="Repeating decimal">repeating decimal</a> in any base b is equivalent to modular multiplication of b modulo the denominator. For example, for decimal, b = 10.

In music, arithmetic modulo 12 is used in the consideration of the system of <a href="Twelve-tone_equal_temperament" class="mw-redirect" title="Twelve-tone equal temperament">twelve-tone equal temperament</a>, where <a href="Octave" title="Octave">octave</a> and <a href="Enharmonic" class="mw-redirect" title="Enharmonic">enharmonic</a> equivalency occurs (that is, pitches in a 1:2 or 2:1 ratio are equivalent, and C-<a href="Sharp_(music)" title="Sharp (music)">sharp</a> is considered the same as D-<a href="Flat_(music)" title="Flat (music)">flat</a>).

The method of <a href="Casting_out_nines" title="Casting out nines">casting out nines</a> offers a quick check of decimal arithmetic computations performed by hand. It is based on modular arithmetic modulo 9, and specifically on the crucial property that 10 ≡ 1 (mod 9).

Arithmetic modulo 7 is used in algorithms that determine the day of the week for a given date. In particular, <a href="Zeller's_congruence" title="Zeller's congruence">Zeller's congruence</a> and the <a href="Doomsday_algorithm" class="mw-redirect" title="Doomsday algorithm">Doomsday algorithm</a> make heavy use of modulo-7 arithmetic.

More generally, modular arithmetic also has application in disciplines such as <a href="Law" title="Law">law</a> (e.g., <a href="Apportionment_(politics)" title="Apportionment (politics)">apportionment</a>), <a href="Economics" title="Economics">economics</a> (e.g., <a href="Game_theory" title="Game theory">game theory</a>) and other areas of the <a href="Social_science" title="Social science">social sciences</a>, where <a href="Proportional_(fair_division)" class="mw-redirect" title="Proportional (fair division)">proportional</a> division and allocation of resources plays a central part of the analysis.

<div class="mw-heading mw-heading2"><h2 id="Computational_complexity">Computational complexity</h2></div>

Since modular arithmetic has such a wide range of applications, it is important to know how hard it is to solve a system of congruences. A linear system of congruences can be solved in <a href="Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a> with a form of <a href="Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>, for details see <a href="Linear_congruence_theorem" class="mw-redirect" title="Linear congruence theorem">linear congruence theorem</a>. Algorithms, such as <a href="Montgomery_reduction" class="mw-redirect" title="Montgomery reduction">Montgomery reduction</a>, also exist to allow simple arithmetic operations, such as multiplication and <a href="Modular_exponentiation" title="Modular exponentiation">exponentiation modulo m</a>, to be performed efficiently on large numbers.

Some operations, like finding a <a href="Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a> or a <a href="Quadratic_congruences" class="mw-redirect" title="Quadratic congruences">quadratic congruence</a> appear to be as hard as <a href="Integer_factorization" title="Integer factorization">integer factorization</a> and thus are a starting point for <a href="Cryptography" title="Cryptography">cryptographic algorithms</a> and <a href="Encryption" title="Encryption">encryption</a>. These problems might be <a href="NP-intermediate" title="NP-intermediate">NP-intermediate</a>.

Solving a system of non-linear modular arithmetic equations is <a href="NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a>.<a href="#cite_note-9">[9]</a>

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<ul><li><a href="Boolean_ring" title="Boolean ring">Boolean ring</a></li>

<li><a href="Circular_buffer" title="Circular buffer">Circular buffer</a></li>

<li><a href="Division_(mathematics)" title="Division (mathematics)">Division (mathematics)</a></li>

<li><a href="Finite_field" title="Finite field">Finite field</a></li>

<li><a href="Legendre_symbol" title="Legendre symbol">Legendre symbol</a></li>

<li><a href="Modular_exponentiation" title="Modular exponentiation">Modular exponentiation</a></li>

<li><a href="Modulo_(mathematics)" title="Modulo (mathematics)">Modulo (mathematics)</a></li>

<li><a href="Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">Multiplicative group of integers modulo n</a></li>

<li><a href="Pisano_period" title="Pisano period">Pisano period</a> (Fibonacci sequences modulo n)</li>

<li><a href="Primitive_root_modulo_n" title="Primitive root modulo n">Primitive root modulo n</a></li>

<li><a href="Quadratic_reciprocity" title="Quadratic reciprocity">Quadratic reciprocity</a></li>

<li><a href="Quadratic_residue" title="Quadratic residue">Quadratic residue</a></li>

<li><a href="Rational_reconstruction_(mathematics)" title="Rational reconstruction (mathematics)">Rational reconstruction (mathematics)</a></li>

<li><a href="Reduced_residue_system" title="Reduced residue system">Reduced residue system</a></li>

<li><a href="Serial_number_arithmetic" title="Serial number arithmetic">Serial number arithmetic</a> (a special case of modular arithmetic)</li>

<li><a href="Two-element_Boolean_algebra" title="Two-element Boolean algebra">Two-element Boolean algebra</a></li>

<li>Topics relating to the group theory behind modular arithmetic:

<ul><li><a href="Cyclic_group" title="Cyclic group">Cyclic group</a></li>

<li><a href="Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">Multiplicative group of integers modulo n</a></li></ul></li>

<li>Other important theorems relating to modular arithmetic:

<ul><li><a href="Carmichael_function" title="Carmichael function">Carmichael's theorem</a></li>

<li><a href="Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a></li>

<li><a href="Euler's_theorem" title="Euler's theorem">Euler's theorem</a></li>

<li><a href="Fermat's_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a> (a special case of Euler's theorem)</li>

<li><a href="Lagrange's_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li>

<li><a href="Thue's_lemma" title="Thue's lemma">Thue's lemma</a></li></ul></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFSandor_LehoczkyRichard_Rusczky2006" class="citation book cs1">Sandor Lehoczky; Richard Rusczky (2006). David Patrick (ed.). the Art of Problem Solving. Vol. 1 (7 ed.). AoPS Incorporated. p. 44. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0977304566</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ModularArithmetic.html">"Modular Arithmetic"</a>. Wolfram MathWorld. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230714132828/https://mathworld.wolfram.com/ModularArithmetic.html">Archived</a> from the original on 2023-07-14. Retrieved 2020-08-12.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFPettofrezzoByrkit1970">Pettofrezzo & Byrkit (1970</a>, p. 90)

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFLong1972">Long (1972</a>, p. 78)

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFLong1972">Long (1972</a>, p. 85)

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Book%3A_Introduction_to_Algebraic_Structures_(Denton)/02%3A_Groups_I/2.03%3A_Integers_Modulo_n">"2.3: Integers Modulo n"</a>. Mathematics LibreTexts. 2013-11-16. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210419035455/https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Book%3A_Introduction_to_Algebraic_Structures_(Denton)/02%3A_Groups_I/2.03%3A_Integers_Modulo_n">Archived</a> from the original on 2021-04-19. Retrieved 2020-08-12.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Sengadir T., <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nglisrt9IewC&pg=PA293&dq=%22Zn+is+generated+by+1%22">Discrete Mathematics and Combinatorics</a>, p. 293, at <a href="Google_Books" title="Google Books">Google Books</a>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://rosettacode.org/wiki/Euler%27s_sum_of_powers_conjecture#QL_SuperBASIC">"Euler's sum of powers conjecture"</a>. rosettacode.org. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230326025754/https://rosettacode.org/wiki/Euler%27s_sum_of_powers_conjecture#QL_SuperBASIC">Archived</a> from the original on 2023-03-26. Retrieved 2020-11-11.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFGareyJohnson1979" class="citation book cs1">Garey, M. R.; Johnson, D. S. (1979). <a rel="nofollow" class="external text" href="https://archive.org/details/computersintract0000gare">Computers and Intractability, a Guide to the Theory of NP-Completeness</a>. W. H. Freeman. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0716710447</bdi>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>John L. Berggren. <a rel="nofollow" class="external text" href="https://www.britannica.com/EBchecked/topic/920687/modular-arithmetic">"modular arithmetic"</a>. <a href="Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a>.</li>

<li><cite id="CITEREFApostol1976" class="citation cs2"><a href="Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90163-3</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a></cite>. See in particular chapters 5 and 6 for a review of basic modular arithmetic.</li>

<li>Maarten Bullynck "<a rel="nofollow" class="external text" href="https://web.archive.org/web/20131102014013/http://www.kuttaka.org/Gauss_Modular.pdf">Modular Arithmetic before C.F. Gauss. Systematisations and discussions on remainder problems in 18th-century Germany</a>"</li>

<li><a href="Thomas_H._Cormen" title="Thomas H. Cormen">Thomas H. Cormen</a>, <a href="Charles_E._Leiserson" title="Charles E. Leiserson">Charles E. Leiserson</a>, <a href="Ronald_L._Rivest" class="mw-redirect" title="Ronald L. Rivest">Ronald L. Rivest</a>, and <a href="Clifford_Stein" title="Clifford Stein">Clifford Stein</a>. <a href="Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a>, Second Edition. MIT Press and McGraw-Hill, 2001. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-03293-7</bdi>. Section 31.3: Modular arithmetic, pp. 862–868.</li>

<li><a rel="nofollow" class="external text" href="http://genealogy.math.ndsu.nodak.edu/id.php?id=3545">Anthony Gioia</a>, Number Theory, an Introduction Reprint (2001) Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-41449-3</bdi>.</li>

<li><cite id="CITEREFLong1972" class="citation book cs1">Long, Calvin T. (1972). Elementary Introduction to Number Theory (2nd ed.). Lexington: <a href="D._C._Heath_and_Company" title="D. C. Heath and Company">D. C. Heath and Company</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/77171950">77171950</a>.</cite></li>

<li><cite id="CITEREFPettofrezzoByrkit1970" class="citation book cs1">Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970). <a rel="nofollow" class="external text" href="https://archive.org/details/elementsofnumber0000pett">Elements of Number Theory</a>. Englewood Cliffs: <a href="Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780132683005</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/71081766">71081766</a>.</cite></li>

<li><cite id="CITEREFSengadir2009" class="citation book cs1">Sengadir, T. (2009). Discrete Mathematics and Combinatorics. Chennai, India: Pearson Education India. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-81-317-1405-8</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/778356123">778356123</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Congruence">"Congruence"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li>In this <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060101075602/http://britton.disted.camosun.bc.ca/modart/jbmodart.htm">modular art</a> article, one can learn more about applications of modular arithmetic in art.</li>

<li>An <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160220061222/http://mersennewiki.org/index.php/Modular_arithmetic">article</a> on modular arithmetic on the GIMPS wiki</li>

<li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/blue/Modulo.shtml">Modular Arithmetic and patterns in addition and multiplication tables</a></li></ul>

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</style><div id="Number_theory52" style="font-size:114%;margin:0 4em"><a href="Number_theory" title="Number theory">Number theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a> (<a href="Class_field_theory" title="Class field theory">class field theory</a>, <a href="Non-abelian_class_field_theory" title="Non-abelian class field theory">non-abelian class field theory</a>, <a href="Iwasawa_theory" title="Iwasawa theory">Iwasawa theory</a>, <a href="Iwasawa%E2%80%93Tate_theory" class="mw-redirect" title="Iwasawa–Tate theory">Iwasawa–Tate theory</a>, <a href="Kummer_theory" title="Kummer theory">Kummer theory</a>)</li>

<li><a href="Analytic_number_theory" title="Analytic number theory">Analytic number theory</a> (<a href="L-function" title="L-function">analytic theory of L-functions</a>, <a href="Probabilistic_number_theory" title="Probabilistic number theory">probabilistic number theory</a>, <a href="Sieve_theory" title="Sieve theory">sieve theory</a>)</li>

<li><a href="Geometry_of_numbers" title="Geometry of numbers">Geometric number theory</a></li>

<li><a href="Computational_number_theory" title="Computational number theory">Computational number theory</a></li>

<li><a href="Transcendental_number_theory" title="Transcendental number theory">Transcendental number theory</a></li>

<li><a href="Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a> (<a href="Arakelov_theory" title="Arakelov theory">Arakelov theory</a>, <a href="Hodge%E2%80%93Arakelov_theory" title="Hodge–Arakelov theory">Hodge–Arakelov theory</a>)</li>

<li><a href="Arithmetic_combinatorics" title="Arithmetic combinatorics">Arithmetic combinatorics</a> (<a href="Additive_number_theory" title="Additive number theory">additive number theory</a>)</li>

<li><a href="Arithmetic_geometry" title="Arithmetic geometry">Arithmetic geometry</a> (<a href="Anabelian_geometry" title="Anabelian geometry">anabelian geometry</a>, <a href="P-adic_Hodge_theory" title="P-adic Hodge theory">p-adic Hodge theory</a>)</li>

<li><a href="Arithmetic_topology" title="Arithmetic topology">Arithmetic topology</a></li>

<li><a href="Arithmetic_dynamics" title="Arithmetic dynamics">Arithmetic dynamics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Number" title="Number">Numbers</a></li>

<li><a href="Natural_number" title="Natural number">Natural numbers</a></li>

<li><a href="1" title="1">Unity</a></li>

<li><a href="Prime_number" title="Prime number">Prime numbers</a></li>

<li><a href="Composite_number" title="Composite number">Composite numbers</a></li>

<li><a href="Rational_number" title="Rational number">Rational numbers</a></li>

<li><a href="Irrational_number" title="Irrational number">Irrational numbers</a></li>

<li><a href="Algebraic_number" title="Algebraic number">Algebraic numbers</a></li>

<li><a href="Transcendental_number" title="Transcendental number">Transcendental numbers</a></li>

<li><a href="P-adic_number" title="P-adic number">p-adic numbers</a> (<a href="P-adic_analysis" title="P-adic analysis">p-adic analysis</a>)</li>

<li><a href="Arithmetic" title="Arithmetic">Arithmetic</a></li>

<li><a href="Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a></li>

<li><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Advanced concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Quadratic_form" title="Quadratic form">Quadratic forms</a></li>

<li><a href="Modular_form" title="Modular form">Modular forms</a></li>

<li><a href="L-function" title="L-function">L-functions</a></li>

<li><a href="Diophantine_equation" title="Diophantine equation">Diophantine equations</a></li>

<li><a href="Diophantine_approximation" title="Diophantine approximation">Diophantine approximation</a></li>

<li><a href="Irrationality_measure" title="Irrationality measure">Irrationality measure</a></li>

<li><a href="Simple_continued_fraction" title="Simple continued fraction">Simple continued fractions</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li> Category</li>

<li> <a href="List_of_number_theory_topics" title="List of number theory topics">List of topics</a></li>

<li> <a href="List_of_recreational_number_theory_topics" title="List of recreational number theory topics">List of recreational topics</a></li>

<li> <a href="https://en.wikibooks.org/wiki/Number_Theory" class="extiw external" title="wikibooks:Number Theory">Wikibook</a></li>

<li> <a href="https://en.wikiversity.org/wiki/Number_Theory" class="extiw external" title="wikiversity:Number Theory">Wikiversity</a></li></ul>

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