By the end of this lesson, you will know all the factors of 96 and learn the easiest ways to find them.

But before we dive into how to figure them out, do you know what a factor is?

A factor is a number that goes into another number without leaving a remainder. For example, 5 is a **factor** of 30, because it goes into 30 perfectly!

Finding factors is easy once you know how to, and the easiest way to do it is to find them in pairs.

96’s Factors

Pairs

Primes

Factorizing 96

How to Find 96’s Factors

Prime Factorization

Method 1: Tree

Method 2: Division

Isn’t 96 Interesting?

To Sum Up (Pun Intended!)

## All Factors of 96

The factors of 96 are **1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96**.

### Factor Pairs for 96

Factors can be written as pairs of numbers that make the original number when multiplied together.

For example **1×96=96**.

1 and 96 are both factors, so the factor pair here is **(1, 96)**.

All the factor pairs for 96 are:

(2, 48)

(3, 32)

(4, 24)

(6, 16)

(8, 12)

In total, 96 has 12 factors. You could also find the negatives:

(-2, -48)

(-3, -32)

(-4, -24)

(-6, -16)

(-8, -12)

Fun Fact: 96 is one of the refactorable numbers!

A **refactorable** number is divisible by its total number of divisors.

96 is divisible by 12, so it is a refactorable number.

### Prime Factors of 96

A prime number is one that can only be divided by 1 and itself.

For example:

- 10 is not a prime number because it can be divided by 2 and 5.
- 13 is a prime number because it can only be divided by 1 and itself

**Prime factors** of a number are the divisors that are divisible only by themselves and 1.

Look at 96’s divisors again:

2 cannot be divided by anything other than 1 and itself, so it is a prime factor of 96.

3 cannot be divided by anything other than 1 and itself, so it is also a prime factor of 96.

So, 96 has two prime factors: 2 and 3.

## Factorizing 96

In this section, you will learn how to find all the factors that you came across in the first sections.

### How to Find the Factors of 96

You already know all of 96’s, but how exactly do you work these out on your own?

One way to find them is to use division.

This means dividing 96 by each number from 1 to 96 and checking if there is a remainder.

If 96 has no remainder when it is divided by a number, then it is a factor. The numbers which do have a remainder are not included as factors.

Each time you find a factor, you will also have found its factor pair!

For example:

So, both the divisor (2) and the answer (48) are factors!

To find the factor pairs of 96, we need to start at 1…

There is no remainder, so the factor pair is 1 and 96.

We now need to check between 2 and 48.

There is no remainder, so the pair is 2 and 48.

There is no remainder, so 3 and 32 are both factors. We now need to check between 3 and 32.

There is no remainder, so the pair is 4 and 24. Then we need to check between 4 and 24.

96 has a remainder when divided by 5 so 5 is not a factor.

There is no remainder, so 6 and 16 are both factors. We need to check between 6 and 16.

So, 7 is not a factor as there is a remainder.

So, 8 and 12 are both factors.

Continuing with this process, we find no factors between 8 and 12, and so have found all the factors of 96.

The negative factors can also be included, so every factor pair takes the negative of each number as a factor.

In total, 96 has the factors: **1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96, -1, -2, -3, -4, -6, -8, -12, -16, -24, -32, -48, -96**.

Here is a table to help you spot factors:

Factor | How to spot the factor |
---|---|

1 | Every whole number has a factor of 1 |

2 | Every even number has a factor of 2 |

3 | A number has a factor of 3 if its digits add up to a number that can be divided by 3 |

4 | A number with a factor of 4 must be even and if the right two digits can be divided by 4 or 00, it has a factor of 4 |

5 | Any number with the last digit 0 or 5 has a factor of 5. |

6 | Any number that its digits sum up to a multiple of 3, and is even, will have a factor of 6 |

7 | 7’s divisibility rule is a bit more complex |

8 | A number with last 3 digits which are a multiple of 8 will have a factor of 8 |

9 | If the sum of a number’s digits is 9, then it has a factor of 9 |

10 | Any number with the last digit 0 has a factor of 10 |

These are so useful in fact, that we made a printable table and worksheet for the divisibility rules from 2 to 15, for you to keep with you. Just click or tap the images below to download them!

Here are three to do yourself. Get the practice, so you remember this for any test!

a) 1, 3, 5, 9, 15, 45, -1, -3, -5, -9, -15, -45

b) 1, 2, 31, 62, -1, -2, -31, -62

c) 1, 2, 3, 6, 9, 18, -1, -2, -3, -6, -9, -18

### Prime Factorization of 96

Prime factorization is finding the unique set of prime numbers that multiply together to make the original number.

By finding all the factors which are divisible only by themselves and 1, you will find that 96 has prime factors 2 and 3.

There are two methods of prime factorization. The easiest way is to use a factor tree.

#### Method One: Factor Tree

A factor tree is a diagram that splits a number up into its factors using branches.

The branches will expand until the end of each branch is a prime factor.

To factorize 96, we need 96 at the start of the tree.

Choosing any factor pair of 96, we can add the pair as branches to the tree. To keep it straightforward, let’s use the pair with the smallest prime, 2.

So, we start with the factor pair **(2, 48)**…

2 can be circled *because it is prime*, but 48 is not, so we need to continue breaking it down. Its smallest prime factor is also 2…

2 is a prime factor, so we can circle it to show it is the end of the branch.

Continuing to break down the factors which aren’t prime by dividing by 2…

3 can be circled as it is a prime. Every branch is now circled, so we have completed the prime factor tree.

We can write 96 as the multiplication of the circled factors.

This should be tidies up to look like this:

^{5}× 3

#### Method Two: Division

The second method involves using the factors of 96 to divide and break it down, starting with the smallest and working up until you get 1 as a result.

Once you have reached 1, you have found the prime factorization needed to express 96.

Start by dividing by 2, the smallest prime factor of 96…

Keep dividing by 2 until you reach the last whole number…

24 ÷ 2 = 12

12 ÷ 2 = 6

6 ÷ 2 = 3

3 ÷ 2 = 1.5

This is not a whole number, so instead, divide by the next smallest prime factor 3.

We have reached 1, so we now have all the prime factors which must be multiplied together to make 96.

We can gather them up and write:

Again, we can simplify this by writing:

^{5}× 3

Notice that we do not include 1, as 1 is not a prime number.

a) 3 × 13

b) 2 × 2 × 17 = 2^{2} × 17

c) 2 × 3 × 3 × 3 = 2 × 3^{3}

## Isn’t 96 Interesting?

The number 96 has some interesting mathematical properties.

96 can be expressed as the sum of odd and even numbers.

- Even numbers:
**96=30+32+34** - Odd numbers:
**96=21+23+25+27**

96 is an **octagonal number**. Octagonal numbers can be arranged in a regular octagon, where the number represents each vertex on the octagons.

There is a town in South Carolina called Ninety-Six, which is a national historic site. It was named this because it was 96 miles from the nearest Cherokee village.

There was a chess version called the Courier game, which was played on a board with 96 squares (8×12) in the 12th century.

The number 96 can be written in Roman numerals as XCVI.

## To Sum Up (Pun Intended!)

A factor divides into another exactly without leaving a remainder. Factors of 96 can be written as factor pairs or prime factors.

To find the factor pairs of 96, you must check each number from 1 to the square root of 96.

To find 96 as a product of prime factors, you need to break the number down using its prime factors. This is prime decomposition.

The two methods of prime decomposition are division and prime factor trees.

How did you get on with the challenges? Let us know or ask us any questions in the comments below!

Here are some of our other factor lessons:

Factors of 27

Factors of 40