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  2. Modulo - Wikipedia

    en.wikipedia.org/wiki/Modulo

    In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation.. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.

  3. Remainder - Wikipedia

    en.wikipedia.org/wiki/Remainder

    For algorithms describing how to calculate the remainder, see division algorithm.) The remainder, as defined above, is called the least positive remainder or simply the remainder . [ 2 ] The integer a is either a multiple of d , or lies in the interval between consecutive multiples of d , namely, q⋅d and ( q + 1) d (for positive q ).

  4. Modular exponentiation - Wikipedia

    en.wikipedia.org/wiki/Modular_exponentiation

    Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = b e mod m. From the definition of division, it follows that 0 ≤ c < m. For example, given b = 5, e = 3 and m = 13, dividing 5 3 = 125 by 13 leaves a remainder of c = 8.

  5. Modulo (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Modulo_(mathematics)

    Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.

  6. Modular arithmetic - Wikipedia

    en.wikipedia.org/wiki/Modular_arithmetic

    explicitly showing its relationship with Euclidean division. 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 remainder when divided by m. That is, a = p m + r, b = q m + r, where 0 ≤ r < m is the common remainder.

  7. Modular multiplicative inverse - Wikipedia

    en.wikipedia.org/wiki/Modular_multiplicative_inverse

    The congruence relation, modulo m, partitions the set of integers into m congruence classes. Operations of addition and multiplication can be defined on these m objects in the following way: To either add or multiply two congruence classes, first pick a representative (in any way) from each class, then perform the usual operation for integers on the two representatives and finally take the ...

  8. Division (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Division_(mathematics)

    The division with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely contained in the first number, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of ...

  9. Casting out nines - Wikipedia

    en.wikipedia.org/wiki/Casting_out_nines

    So we can use the remainder from casting out nines to get the remainder of division by three. Casting out ninety nines is done by adding groups of two digits instead just one digit. Since 11·9 = 99, = So we can use the remainder from casting out ninety nines to get the remainder of division by eleven. This is called casting out elevens.

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