What it is
A whole number a is divisible by a whole number b (not 0) when a/b is a whole number, that is, when there is a whole number k with a = b · k. Then b is a divisor (or factor) of a, and a is a multiple of b.
- 12 is divisible by 4, because 12 = 3 · 4. The divisors of 12 are 1, 2, 3, 4, 6 and 12.
- 84 is divisible by 12, because 84 = 7 · 12.
- 0 is divisible by every whole number except 0, because 0 = 5 · 0, and so on.
- 13 is not divisible by 4: 13 = 3 · 4 + 1, so the remainder is 1.
- 5 is not divisible by 10: 5/10 = 0.5 is not a whole number.
- 7 is not divisible by 2: 7 = 3 · 2 + 1.
The remainder of a division is always smaller than the divisor, and a number is divisible exactly when the remainder is 0. Every whole number is divisible by 1 and by itself. A number with just those two divisors is a prime number.
Two rules that get used a lot: if b divides both a and c, then b also divides a + c and a − c. 6 divides 18 and 24, and so it divides 42 = 18 + 24 (42 = 7 · 6). And if b divides a, then b divides every multiple of a.
Why it matters
Divisibility sits under most of what is done with whole numbers:
- Simplifying a fraction works only when numerator and denominator share a divisor. 84/360 can be reduced by 12 because 12 divides both: 84 = 7 · 12 and 360 = 30 · 12, giving 7/30.
- Greatest common divisor and least common multiple are the largest divisor and the smallest multiple two numbers share, see GCD and LCM.
- Prime factorisation splits a number into divisors that cannot be split further, see prime factorisation.
- Remainders and clock arithmetic: two numbers leaving the same remainder on division by n is exactly what modular arithmetic is about.
To avoid dividing there are the divisibility rules: a number is divisible by 3 if its digit sum is (123 has digit sum 6, and 123 = 3 · 41), and divisible by 4 if its last two digits form a number that is (316: 16 = 4 · 4, and 316 = 4 · 79).
The GCD calculator, the LCM calculator and the prime factorization calculator use divisibility step by step.
Common misconceptions
- “0 is not divisible by anything.” It is: 0 = b · 0 for every b, so 0 is divisible by every whole number except 0. It is dividing by 0 that does not work.
- “Divisible by two numbers means divisible by their product.” Not always: 12 is divisible by both 4 and 6 but not by 24, and by both 2 and 4 but not by 8. It holds when the two numbers share no divisor other than 1: a number divisible by both 2 and 3 is divisible by 6.
- “The remainder can be as large as you like.” The remainder is always smaller than the divisor. If it were as large, the divisor would go in once more.
- “Divisor and multiple mean the same.” They are opposites: 4 is a divisor of 12, and 12 is a multiple of 4.
Common questions
What is the difference between divisor, factor and multiple?
Divisor and factor mean the same here: a number that goes into another evenly. A multiple is the larger
number, what you get by multiplying by a whole number. 3 is a divisor and a factor of 12, and 12 is a
multiple of 3 (and of 1, 2, 4, 6 and 12).
How many divisors does a number have?
It varies: 13 has two (1 and 13), 12 has six (1, 2, 3, 4, 6, 12) and 36 has nine. Divisors come in pairs that
multiply to the number: 12 = 1 · 12 = 2 · 6 = 3 · 4. A number has an odd number of divisors exactly when it
is a square number.
Can negative numbers be divisible?
The same definition works: −12 is divisible by 4, because −12 = (−3) · 4, and 12 is divisible by −4. Usually
only the positive divisors are talked about.