In mathematics, factors are numbers that divide a given number evenly without leaving a remainder. The factors of 72 are all the numbers that can divide 72 completely. To find the factors of 72, you simply check which numbers, when divided into 72, result in an exact whole number. Understanding the factors of a number like 72 is helpful in various mathematical problems like simplifying fractions, finding greatest common divisors, and more.

## Factors Calculator

## What Are the Factors of 72?

The factors of 72 are the numbers that, when multiplied in pairs, give the product as 72. Let’s begin by finding all the factors of 72 and understanding why each is a factor:

- 1 is a factor of 72 because 72 ÷ 1 = 72
- 2 is a factor of 72 because 72 ÷ 2 = 36
- 3 is a factor of 72 because 72 ÷ 3 = 24
- 4 is a factor of 72 because 72 ÷ 4 = 18
- 6 is a factor of 72 because 72 ÷ 6 = 12
- 8 is a factor of 72 because 72 ÷ 8 = 9
- 9 is a factor of 72 because 72 ÷ 9 = 8
- 12 is a factor of 72 because 72 ÷ 12 = 6
- 18 is a factor of 72 because 72 ÷ 18 = 4
- 24 is a factor of 72 because 72 ÷ 24 = 3
- 36 is a factor of 72 because 72 ÷ 36 = 2
- 72 is a factor of 72 because 72 ÷ 72 = 1

**So, the factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.**

### Factor Pairs of 72

Factor pairs are the numbers that, when multiplied together, result in 72. These pairs of numbers can be positive or negative. Let’s list all the factor pairs:

- 1×72=72
- 2×36=72
- 3×24=72
- 4×18=72
- 6×12=72
- 8×9=72

**Thus, the factor pairs of 72 are:**

(1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9).

For negative factors, the pairs are:

(-1, -72), (-2, -36), (-3, -24), (-4, -18), (-6, -12), (-8, -9).

### Prime Factorization of 72

Prime factorization involves breaking down a number into its prime factors, which are numbers divisible only by 1 and themselves. For 72, we divide it by the smallest prime number (2) and continue dividing until we can’t anymore.

- 72÷2=36
- 36÷2=18
- 18÷2=9
- 9÷3=3
- 3÷3=1

So, the prime factorization of 72 is:

72=2^3×3^2

### Factor Tree of 72

A factor tree helps in visually breaking down a number into its prime factors. Starting with 72, we continuously break it into its factor pairs:

```
72
/ \
2 36
/ \
2 18
/ \
2 9
/ \
3 3
```