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Modular arithmetic

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Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus. Modular arithmetic was introduced by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

A familiar use of modular arithmetic is its use in the 24-hour clock: the arithmetic of time-keeping in which the day runs from midnight to midnight and is divided into 24 hours, numbered from 0 to 23. If the time is 19:00 now — 7 o'clock in the evening — then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 19 + 8 = 27, but this is not the answer because clock time "wraps around" at the end of the day. Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 09:00 the next day, rather than 33:00. Since the hour number starts over when it reaches 24, this is arithmetic modulo 24. It should be noted that in this system 24:00 is not a valid time because this is equal to the 0:00 of the next day, in much the same way 2:60 is not a valid time because it is equal to 3:00.

Time-keeping on a clock gives an example of modular arithmetic.

The congruence relation

Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations of the ring of integers: addition, subtraction, and multiplication. For a fixed modulus n, it is defined as follows.

Two integers a and b are said to be congruent modulo n, if their difference a − b is an integer multiple of n. If this is the case, it is expressed as:

a \equiv b \pmod n.\,

The above mathematical statement is read: "a is congruent to b modulo n".

For example,

38 \equiv 14 \pmod {12}\,

because 38 − 14 = 24, which is a multiple of 12. For positive n and non-negative a and b, congruence of a and b can also be thought of as asserting that these two numbers have the same remainder after dividing by the modulus n. So,

38 \equiv 2 \pmod {12}\,

because, when divided by 12, both numbers give 2 as remainder.

The same rule holds for negative values of a:

-3 \equiv 2 \pmod 5.\,

A remark on the notation: Because it is common to consider several congruence relations for different moduli at the same time, the modulus is incorporated in the notation. In spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been clearer if the notation a n b had been used, instead of the common traditional notation.

The properties that make this relation a congruence relation (respecting addition, subtraction, and multiplication) are the following.

If a_1 \equiv b_1 \pmod n and a_2 \equiv b_2 \pmod n, then:

  • (a_1 + a_2) \equiv (b_1 + b_2) \pmod n\,
  • (a_1 - a_2) \equiv (b_1 - b_2) \pmod n\,
  • (a_1 a_2) \equiv (b_1 b_2) \pmod n\,

The ring of congruence classes

Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by \overline{a}_n, is the set \left\{\ldots, a - 2n, a - n, a, a + n, a + 2n, \ldots \right\}. This set, consisting of the integers congruent to a modulo n, is called the congruence class or residue class of a modulo n. Another notation for this congruence class, which requires that in the context the modulus is known, is \displaystyle [a].

The set of congruence classes modulo n is denoted as \mathbb{Z}/n\mathbb{Z} and defined by:

\mathbb{Z}/n\mathbb{Z} = \left\{ \overline{a}_n | a \in \mathbb{Z}\right\}.

When n ≠ 0, \mathbb{Z}/n\mathbb{Z} has n elements, and can be written as:

\mathbb{Z}/n\mathbb{Z} = \left\{ \overline{0}_n, \overline{1}_n, \overline{2}_n,\ldots, \overline{n-1}_n \right\}.

When n = 0, \mathbb{Z}/n\mathbb{Z} does not have zero elements; rather, it is isomorphic to \mathbb{Z}, since \overline{a}_0 = \left\{a\right\}.

We can define addition, subtraction, and multiplication on \mathbb{Z}/n\mathbb{Z} by the following rules:

  • \overline{a}_n + \overline{b}_n = \overline{a + b}_n
  • \overline{a}_n - \overline{b}_n = \overline{a - b}_n
  • \overline{a}_n \overline{b}_n = \overline{ab}_n.

The verification that this is a proper definition uses the properties given before.

In this way, \mathbb{Z}/n\mathbb{Z} becomes a commutative ring. For example, in the ring \mathbb{Z}/24\mathbb{Z}, we have

\overline{12}_{24} + \overline{21}_{24} = \overline{9}_{24}

as in the arithmetic for the 24-hour clock.

The notation \mathbb{Z}/n\mathbb{Z} is used, because it is the factor ring of \mathbb{Z} by the ideal n\mathbb{Z} containing all integers divisible by n, where 0\mathbb{Z} is the singleton set \left\{0\right\}.

In terms of groups, the residue class \overline{a}_n is the coset of a in the quotient group \mathbb{Z}/n\mathbb{Z}, a cyclic group.

The set \mathbb{Z}/n\mathbb{Z} has a number of important mathematical properties that are foundational to various branches of mathematics.

Rather than excluding the special case n = 0, it is more useful to include \mathbb{Z}/0\mathbb{Z} (which, as mentioned before, is isomorphic to the ring \mathbb{Z} of integers), for example when discussing the characteristic of a ring.


The notion of modular arithmetic is related to that of the remainder in division. The operation of finding the remainder is sometimes referred to as the modulo operation and we may see "2 = 14 (mod 12)". The difference is in the use of congruency, indicated by ≡, and equality indicate by =. Equality implies specifically the "common residue", the least non-negative member of an equivalence class. When working with modular arithmetic, each equivalence class is usually represented by its common residue, for example "38 ≡ 2 (mod 12)" which can be found using long division. It follows that, while it is correct to say "38 ≡ 14 (mod 12)", "2 ≡ 14 (mod 12)" and "2 ≡ 14 (mod 12)", it is incorrect to say "38 = 14 (mod 12)" (with "=" rather than "≡").

Parentheses are sometimes dropped from the expression, e.g. "38 ≡ 14 mod 12" or "2 = 14 mod 12", or placed around the divisor e.g. "38 ≡ 14 mod (12)". Notation such as "38(mod 12)" has also been observed, but is ambiguous without contextual clarification.

The congruence relation is sometimes expressed by using modulo instead of mod, like "38 ≡ 14 (modulo 12)" in computer science. The modulo function in various computer languages typically yield the common residue, for example the statement "y = MOD(38,12);" gives y = 2.


Modular arithmetic is referenced in number theory, group theory, ring theory, knot theory, abstract algebra, cryptography, computer science, chemistry and the visual and musical arts.

It is one of the foundations of number theory, touching on almost every aspect of its study, and provides key examples for group theory, ring theory and abstract algebra.

In cryptography, modular arithmetic directly underpins public key systems such as RSA and Diffie-Hellman, as well as providing finite fields which underlie elliptic curves, and is used in a variety of symmetric key algorithms including AES, IDEA, and RC4.

In computer science, modular arithmetic is often applied in bitwise operations and other operations involving fixed-width, cyclic data structures. The modulo operation, as implemented in many programming languages and calculators, is an application of modular arithmetic that is often used in this context.

In chemistry, the last digit of the CAS registry number (a number which is unique for each chemical compound) is a check digit, which is calculated by taking the last digit of the first two parts of the CAS registry number times 1, the next digit times 2, the next digit times 3 etc., adding all these up and computing the sum modulo 10.

In the visual arts, modular arithmetic can be used to create artistic patterns based on the multiplication and addition tables modulo n (see external link, below).

In music, arithmetic modulo 12 is used in the consideration of the system of twelve-tone equal temperament, where octave and enharmonic equivalency occurs (that is, pitches in a 1∶2 or 2∶1 ratio are equivalent, and C- sharp is considered the same as D- flat).

The method of casting out nines 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).

More generally, modular arithmetic also has application in disciplines such as law (see e.g., apportionment), economics, (see e.g., game theory) and other areas of the social sciences, where proportional division and allocation of resources plays a central part of the analysis.

Some neurologists (see e.g., Oliver Sacks) theorize that so-called autistic savants utilize an "innate" modular arithmetic to compute such complex problems as what day of the week a distant date will fall on.

Computational complexity

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 polynomial time with a form of Gaussian elimination, for details see the linear congruence theorem.

Solving a system of non-linear modular arithmetic equations is NP-complete. For details, see for example M. R. Garey, D. S. Johnson: Computers and Intractability, a Guide to the Theory of NP-Completeness, W. H. Freeman 1979.

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