If the terms

*a*and*b*of a division are non-zero whole numbers, with*a*≥*b*, then*b*is a whole divisor of*a*if and only if the remainder of the division is zero.Similarly, if the terms

*a*and*b*of a division are non-zero integers, with*a*≥*b*, then*b*is an integral divisor of*a*if and only if the remainder of this division is zero. If an integer*d*is a divisor of an integer*N*, then the integer –*d*is also an integral divisor of*N*.### Notation

- The set of (positive) integral divisors of a whole number
*n*is noted as “div(*n*)” and is read as “the set of integral divisors of*n*“.

We note the set of positive or negative integral divisors of an integer in the same way. - The set of whole divisors of 60, noted as div(60), is: div(60) = {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}.
- The set of integral divisors of 12, noted as div(12), is: div(12) = {–12, –6, –4, –3, –2, –1, 1, 2, 3, 4, 6, 12}.

If not otherwise specified in a given context, div(*n*) indicates the set of positive divisors of *n*.

### Example

In the operation 12 ÷ 4 = 3, the number 4 is the integral divisor of 12 because the remainder of this division is zero. The (positive) integral divisors of 12 are {1, 2, 3, 4, 6, 12}.