# Divisor

A divisor is the number that the dividend is divided by in a division problem. The term divisor is also sometimes used as a synonym for factor (discussed below).

Division problems can be presented in a number of different ways, so it is important to be able to recognize the components of a division problem in different formats. In the figure below, the divisor, 3, is shown in purple.

In the problem above, the dividend is 12, and the quotient, or the solution to the division problem, is 4.

We can conceptualize the divisor as the number of groups of objects in a division problem. The dividend is the total number of objects, the divisor is the number of equal groups of objects, and the quotient is the number of objects in each group.

## Divisor (factor) of an integer

In some cases, the term divisor is used as a synonym for a factor. In these cases, the divisor is the number that divides the dividend (an integer) with no remainder. In the example 12 ÷ 3 = 4, 3 qualifies as a divisor based on both definitions, since it divides 12 with no remainder. On the other hand, 12 ÷ 5 fits the definition of the number that divides the dividend, but it is not a factor because it leaves a remainder (12 ÷ 5 = 10 R2).

### Negative divisors

Divisors can also be negative, though in some cases, only positive divisors are considered.

Example

Identify all the divisors of the following integers

1. 4

Divisors of 4: 1, 2, 4, -1 , -2, -4

2. 12

Divisors of 12: 1, 2, 3, 4, 6, 12, -1, -2, -3, -4, -6, -12

3. 20

Divisors of 20: 1, 2, 4, 5, 10, 20, -1, -2, -4, -5, -10, -20

If a positive integer is a divisor, then its negative counterpart will also be a divisor. As such, depending on the context, the negative divisors can be left out.