Divisibility
Divisibility means one integer divides another with no remainder — 12 is divisible by 3 because 12 ÷ 3 = 4 exactly. Divisibility rules and prime factorization make these checks fast without long division.
Divisibility describes when one integer divides another evenly. Formally, an integer a is divisible by a nonzero integer b if there is an integer k with a = bk — equivalently, dividing a by b leaves remainder zero. In that case b is called a factor (or divisor) of a, and a is a multiple of b.
Divisibility rules let you test common factors quickly. A number is divisible by 2 if its last digit is even; by 3 if its digit sum is divisible by 3; by 4 if its last two digits form a number divisible by 4; by 5 if it ends in 0 or 5; by 6 if it passes the tests for 2 and 3; by 9 if its digit sum is divisible by 9; and by 10 if it ends in 0. For example, 4,572 is divisible by 4 because 72 is, and 8,316 is divisible by 9 because 8 + 3 + 1 + 6 = 18.
Divisibility is the backbone of elementary number theory. Prime factorization expresses any integer as a product of primes, which in turn determines every divisor the number has, its greatest common factor with other numbers, and its least common multiple. Modular arithmetic generalizes divisibility by tracking remainders instead of just checking for zero.
Competition and licensure math exams lean on divisibility constantly: AMC 8 and AMC 12 number theory problems ask for digit-based divisibility reasoning, counting divisors, and remainder arguments, while Praxis Core Math tests factors, multiples, and divisibility rules as properties of whole numbers.
Key takeaways
- An integer a is divisible by b when a = bk for some integer k — division leaves no remainder.
- Digit-based rules give fast tests for divisibility by 2, 3, 4, 5, 6, 9, and 10.
- Prime factorization determines all of a number's divisors, its GCF with other numbers, and its LCM.
- Modular arithmetic extends divisibility thinking from "remainder zero" to remainders in general.
- AMC 8, AMC 12, and Praxis Core Math all test divisibility through digit rules, factors, and remainder reasoning.