# 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).

## Divisor meaning

A divisor is the number in a division problem that either divides the dividend completely, or with a remainder.

### Parts of a division problem

The three main parts of a division problem are: divisor, quotient, dividend.

- Dividend - the dividend is the number that is being divided.
- Divisor - the divisor is the number that the dividend is being divided by.
- Quotient - the quotient is the result of dividing the dividend by the divisor; it is the solution to the division problem.

### Divisor formula

The divisor formula can be written in two ways: with remainder or without remainder. The divisor formula without remainder is:

divisor = dividend ÷ quotient

The divisor formula with remainder is:

divisor = (dividend - remainder) ÷ quotient

Below are some divisor examples.

Examples

1. Find the divisor given a dividend of 36 and quotient of 3.

Since there is no remainder, we use the divisor formula without remainder:

divisor = 36 ÷ 3 = 12

Thus, the divisor is 12.

2. Find the divisor given a dividend of 50, quotient of 12, and remainder of 2.

Since there is a remainder, we use the divisor formula with remainder:

divisor = (50 - 2) ÷ 12 = 4

Thus, the divisor is 4.

## How to write a division problem

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 vs factor

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 with no remainder. Note that all factors are divisors, but not all divisors are factors.

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).

### Can a divisor be negative

Divisors can be negative, but usually, when discussing divisors, only positive divisors are considered.

### Find factors of a number

Below are some examples of how to find factors of a number, including using negative divisors.

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.

## Dividend vs divisor

A dividend and divisor are both part of a division problem. In a division problem, a dividend is the number being divided, and a divisor is the number that the dividend is divided by.

## Table of divisors

The following table shows the positive divisors for the integers (n) 1 through 15.

n | divisors |
---|---|

1 | 1 |

2 | 1, 2 |

3 | 1, 3 |

4 | 1, 2, 4 |

5 | 1, 5 |

6 | 1, 2, 3, 6 |

7 | 1, 7 |

8 | 1, 2, 4, 8 |

9 | 1, 3, 9 |

10 | 1, 2, 5, 10 |

11 | 1, 11 |

12 | 1, 2, 3, 4, 6, 12 |

13 | 1, 13 |

14 | 1, 2, 7, 14 |

15 | 1, 3, 5, 15 |

Notice from the table that there are some special cases of divisors. For example:

- 1 is always a divisor.
- A number is always a divisor of itself.