Divisibility rule

A shortcut for deciding whether an integer is divisible by a given integer, using its digits or other properties instead of carrying out division.

Modular arithmetic: Arithmetic in which numbers are grouped by their remainders upon division by a fixed positive integer. Divisibility rules work by determining whether a number’s remainder modulo the divisor is zero.
Divisibility by 2: The property of an integer being a multiple of two. In decimal notation, the final digit alone determines whether a number is divisible by two.
Integer: A whole number that is zero, positive, or negative. Divisibility rules decide whether one integer is an exact multiple of another.
Long division: A written algorithm for dividing numbers one digit at a time. Unlike a divisibility rule, long division computes a quotient and remainder directly.
Place-value notation: A numeral system in which a digit’s value depends on its position and the system’s base. Rules for decimal digits follow from powers of ten and their remainders modulo the divisor.
Divisibility by 3: The property of an integer being a multiple of three. A decimal number is divisible by three exactly when its digit sum is divisible by three.
Divisibility: The relation in which one integer is a factor of another with no remainder. A rule is a shortcut for deciding this exact-multiple relation.
Trial division: A primality-testing method that checks candidate divisors up to the square root of a number. A divisibility rule can speed individual checks, while trial division systematically tests candidates.
Congruence relation: A relation stating that two integers have the same remainder when divided by a specified modulus. Replacing powers of ten with congruent remainders makes many digit tests possible.
Divisibility by 9: The property of an integer being a multiple of nine. The digit-sum test works because every power of ten is congruent to one modulo nine.

Linked from 3 pages

Perfect squareRelated: Remainder patterns provide quick tests for whether an integer could be a square.