Divisors
A divisor of an integer is a number that divides it evenly. The terms divisor and factor mean the same thing in whole-number arithmetic.
Definition
An integer d is a divisor of n if there exists an integer k such that n = d × k. Equivalently, n mod d = 0 (n divided by d leaves no remainder).
Divisors come in pairs: if d divides n, then n/d also divides n. For example, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Counting Divisors
The number of positive divisors of n can be found from its prime factorization. If n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ, then the divisor count is (a₁ + 1)(a₂ + 1)…(aₖ + 1).
For 12 = 2² × 3¹, the divisor count is (2 + 1)(1 + 1) = 6, matching the six divisors 1, 2, 3, 4, 6, and 12.
Divisors in Practice
Divisors are used to find the greatest common divisor (GCD) and least common multiple (LCM) of two numbers.
The GCD is the largest integer that divides both numbers, while the LCM is the smallest positive integer divisible by both.
Examples
- The positive divisors of 15 are 1, 3, 5, and 15.
- 6 is a divisor of 42 because 42 = 6 × 7.
- The number 13 has exactly two divisors: 1 and 13.
FAQ
Are divisor and factor the same?
Yes, for positive integers. Both terms refer to a number that divides another evenly. "Factor" is more common in elementary math; "divisor" appears more often in number theory.
Does every number divide itself?
Yes. For any nonzero integer n, n = n × 1, so n is always a divisor of itself. The number 1 divides every integer.