In this lesson, you’ll learn what the factors of 36 are and how to find them.

The factors of 36 are all the numbers that go into 36 without leaving a remainder.

By the time you’ve finished, you’ll be able to find the factors of any number you want!

You will also learn about factor pairs, prime factors, and factor trees, which can help find a number’s prime composition.

36’s Factors

Pairs

Primes

Factorizing

How to Find 36’s Factors

Prime Factorization

Isn’t 36 Interesting?

To Sum Up (Pun Intended!)

## All Factors of 36

The factors of 36 are **1, 2, 3, 4, 6, 9, 12, 18, and 36**!

### What are the Factor Pairs of 36?

A **factor pair** is a pair of whole numbers that multiply together to give a particular number.

So, a factor pair of 36 is a pair of whole numbers that multiply together to make 36!

Instead of saying ‘whole numbers,’ mathematicians use the word **integer**.

(2, 18) (-2, -18)

(3, 12) (-3, -12)

(4, 9) (-4, -9)

(6, 6) (-6, -6)

For every factor pair of positive numbers, there is also a pair of negative numbers. This is because when two negative numbers multiply together, the answer is always a positive number.

Using the factor pairs, we can write out all the factors of 36 in order like this:

This is every factor of 36. Sometimes they are called **divisors**.

Divisors of a number are those which divide it exactly, without leaving a fraction or a remainder.

### Prime Factors of 36

Do you recognize this sequence?

These are the **prime numbers**: all the whole numbers that can only be divided by 1 and themselves.

The **prime factors** of a number are those that are also prime.

So for 36, its prime factors will be the numbers that appear in both the lists above. For 36 there are only 2:

## Factorizing 36

### How to Find the Factors of 36

If you want to find all the factors without using a calculator, you can use a few mental math tricks called divisibility tests.

These tests let you tell at a glance whether or not a number can be divided exactly.

They 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!

Once you have one factor, you can easily find the other factor.

To find all the factors of a number, you start checking if 36 is divisible by 1, 2, 3, 4, 5, etc., getting the factor pairs until you meet in the middle—when the next factor to test is one you’ve already found as the other half of a factor pair.

This method makes sure that you won’t miss any, as long as you get the divisibility test correct!

Let’s have a look at some of these tests:

**1**: No test necessary! All numbers are divisible by 1.

For 36, this gives the factor pair **(1,36)**

**2**: the last digit is 0, 2, 4, 6, or 8.

If the last digit is even, then the whole number is even. 36 ends in 6, so it has a factor of 2. Then you can work out **36÷2=18** to get the factor pair **(2,18)**.

**3**: the digits’ sum is a multiple of 3.

In the case of 36, the digit sum is **3+6=9=3×3**, and so 36 has a factor of 3, giving the factor pair **(3,12)**.

The same method can be applied again and again to deal with large numbers:

Show that 94,247,778 is a multiple of 3.

Start by adding the digits together.

Repeat this process.

**12=3×4** is a multiple of 3, meaning so is 48, and finally so is 94,247,778. You don’t actually need to find the other part of the factor pair to show that 3 is a factor.

This shows how useful divisibility tests can be—the number in this example is large, but the math needed for the test can be done in your head!

**4**: last 2 digits are a multiple of 4.

This isn’t really helpful for 36, since it is only 2 digits long anyway, but it is useful with large numbers.

For example, 1,000,024 is a multiple of 4 since the last two digits are **24=4×6**, which is a multiple of 4.

However, something can still be done for 36: since **4=2×2**, being divisible by 4 is the same as being ‘divisible by 2 twice’, meaning the second half of the factor pair with 2, which was 18, must also have a factor of 2 for 36 to have a multiple of 4.

Since 18 ends in 8, it does have a factor of 2—it is divisible 4.

so the factor pair is **(4,8)**

**5**: the last digit is either 0 or 5.

Probably the simplest test of them all, it immediately shows that 5 is *not *a factor of 36.

**6**: divisibly by 2 and 3.

This is similar to the method shown for divisibility by 4: since **6=2×3**, any number that’s divisible by 6 must also be divisible by both 2 and 3.

Looking back at the earlier tests, this is indeed the case for 36, and so 6 is a repeated factor of 36, giving the factor pair **(6,6)**

And you can stop there because 6, which is the largest factor tested so far, also appears in the other half of the factor pair **(6,6)**.

They are 1, 2, 3, 5, 6, 10, 15 and 30.

### The Prime Factorization of 36

Remember that the prime factors of 36 were 2 and 3.

It turns out that any number can be written *uniquely *as a product of its prime factors. This is called its **prime factorization**.

- First, split up the number into any factor pair, connected by ‘branches’ to the starting number.
- If a factor is
*prime*, circle it—the branch stops there. Otherwise, for all the non-circled numbers, repeat the process of splitting each number into its own pair. - Keep doing this until all the branches end in circled prime factors. Then these circled numbers are the prime factorization.

Let’s look at 36 as an example:

The factor tree shows that the prime factorization of 36 is:

This should be simplified to look like this:

^{2}× 3

^{2}

Because the prime factorization is unique, it doesn’t matter how you choose to split up the numbers when making the factor tree. You’ll always be left with the same combination of prime factors at the end!

The prime factorization of 30 is 2 × 3 × 5

## Isn’t 36 Interesting?

Here are some interesting facts about the number 36 that you might not know:

- In French, 36 is ‘
*trente-six*,’ and it is used as a placeholder number, meaning it’s used in place of a large unknown number, just like ‘a dozen’ or ‘umpteen’ is used in English. - French isn’t the only language with unique and interesting ways of expressing large unknown numbers in speech:
- In Sweden they say ‘
*femtioelva*‘ (fifty-eleven) - In Spain, ‘
*veinticatorce*‘ (twenty-fourteen)

- In Sweden they say ‘

What is used for placeholder numbers in a language that you speak? We’d love to see what variety of different numbers are used!

From a mathematical perspective, 36 is also an interesting number:

- As we’ve already seen,
**36=6**, so it’s a^{2}**square number**: 36 identical blocks can be arranged to form a square with 6 blocks along each side. - Not only is it a square number, but it’s also a
**triangular number**, so the 36 blocks from before can also be arranged to form a triangle. - A number that is both a square number and a triangular number is called a
**square triangular number**, and 36 is the smallest example… if you exclude 1, because 1 makes a boring square and triangle! - Square triangular numbers are quite rare: the next one after 36 is 1,225, then they keep getting larger and rarer:1

36

1,225

41,616

1,413,721

48,024,900

…

## To Sum Up (Pun Intended!)

We found the factors of 36 by using **divisibility tests** to find different factor pairs. They are:

Then we used **factor trees** to find the unique prime factorization of 36:

^{2}3

^{2}

If you’re still unsure about either of these two methods, have a go at them yourself—choose a number, try and find its factors using divisibility tests, and then use a factor tree to find its prime factorization.

Leave us a comment below if you’ve got a question about anything in the article, or if you’ve got an interesting fact about 36 that you think should have been mentioned—we’d love to hear it!