Given two integers d and a, where d is nonzero, d is said to divide a, or d is said to be a divisor of a, if and only if there is an integer k such that dk = a. For example, 3 divides 6 because 3 · 2 = 6. Here 3 and 6 play the roles of d and a, while 2 plays the role of k. Though any number divides itself (as does its negative), it is said not to be a proper divisor. The number 0 is not considered to be a divisor of any integer.
- 6 is a divisor of 24 since . (We stress that "6 divides 24" and "6 is a divisor of 24" mean the same thing.)
- 5 divides 0 because . In fact, every integer except zero divides zero.
- 7 is a divisor of 49 since .
- 7 divides 7 since .
- 1 divides 5 because .
- −3 divides 9 because
- −4 divides −16 because
- 2 does not divide 9 because there is no integer k such that . Since 2 is not a divisor of 9, 9 is said to be an odd integer, or simply an odd number.
- Given that d is non zero, the number k such that dk = a is unique and is called the exact quotient of a by d, denoted a/d.
- 0 can never be a divisor of any number. It is true that 0 · k = 0 for any k, however, the quotient 0/0 is not defined, as any k would work. This is the reason 0 is excluded from being considered a divisor.
See also Division (arithmetic).
If d is a divisor of a (we also say that d divides a), this fact may be expressed by writing d | a. Similarly, if d does not divide a, we write . For example, 4 | 12 but .
 Related concepts
If d is a divisor of a (d | a), we say a is a multiple of d. For example, since 4 | 12, 12 is a multiple of 4. If both d1 and d2 are divisors of a, we say a is a common multiple of d1 and d2. Ignoring the sign (i.e., only considering nonnegative integers), there is a unique greatest common divisor of any two integers a and b written or, more commonly, (a,b). The greatest common divisor of 12 and 8 is 4, the greatest common divisor of 15 and 16 is 1. Two numbers with a greatest common divisor of 1 are said to be relatively prime. Complementary to the notion of greatest common divisor is least common multiple. The least common multiple of a and b is the smallest (positive) integer m such that a | m and b | m. Thus, the least common multiple of 12 and 9 is 36 (written [12,9] = 36).
 Abstract divisors
 Further reading
- Scharlau, Winfried; Opolka, Hans (1985). From Fermat to Minkowski: Lectures on the Theory of Numbers and its Historical Development. Springer-Verlag. ISBN 0-387-90942-7.
|Some content on this page may previously have appeared on Citizendium.|