# Integral Divisor

## Integral Divisor

If the terms a and b of a division are non-zero whole numbers, with ab, 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 ab, 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}.