Remainder

A remainder is the integer left after dividing one integer by a nonzero integer, when the division is not exact. In Euclidean division, it is chosen to be nonnegative and smaller than the divisor’s absolute value.

Euclidean division: A division rule expressing an integer as a multiple of a divisor plus a bounded remainder. It states the quotient-and-remainder equation that defines this concept.
Integer: A whole number that is zero, positive, or negative. Remainders in this sense arise from dividing integers.
Euclidean algorithm: A procedure for finding the greatest common divisor by repeatedly dividing and replacing numbers with remainders. Its successive remainders turn division into an efficient computational method.
Least nonnegative residue: The unique representative from zero through one less than a positive modulus in an integer’s congruence class. It is the standard nonnegative remainder convention for a positive divisor.
Division algorithm: The theorem that integers divide with a unique quotient and a remainder in a specified range. It guarantees that every integer division produces exactly one such remainder.
Divisor: An integer that divides another integer without leaving a fractional part. Its magnitude sets the allowed range of Euclidean remainders.
Divisibility rule: A shortcut for deciding whether one integer divides another using properties of its digits or representation. Many rules infer divisibility from a number’s remainder in a chosen base.
Signed remainder: A remainder convention in which the result may be negative and typically follows the dividend’s sign. It differs from Euclidean remainder for some negative dividends.
Quotient: The result of division, measuring how many times one quantity is contained in another. Together with the divisor, it determines the part removed before the remainder is left.
Multiple: A number obtained by multiplying an integer by another integer. The quotient times the divisor is the multiple subtracted from the dividend.

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