The factors of 50 are numbers that multiply into 50 without leaving a remainder.

For example, 50 can be divided by 10, so 10 is a factor!

Factors and multiples are connected to each other. In the example of:

10 and 5 are both factors of the multiple 50.

But what are the rest of them, and how do we find them?

This lesson will cover those questions, and more, including how to find factor pairs, and the prime factorization of 50 too.

50’s Factors

Pairs

Primes

Factorizing 50

Prime Factorization

How to Find Factors

Method 1: Using Divisibility Rules

Method 2: Using Prime Factorization

Isn’t 50 Interesting?

To Sum Up (Pun Intended!)

## All the Factors of 50

The factors of 50 are **1, 2, 5, 10, 25, and 50!**

### What are the Factor Pairs of 50?

The **factor pairs** of 50 are all pairs of whole numbers that multiply together to give 50.

Here are all the pairs for 50:

Positive | Negative |
---|---|

(1, 50) | (-1, -50) |

(2, 25) | (-2, -25) |

(5, 10) | (-5, -10) |

For each pair of positive numbers that multiply together to make 50, there is also a corresponding pair of negative numbers, because the product of two negative numbers is positive.

From the list of factor pairs above, you get the following list, ordered from smallest to largest:

Since 1 and 50 are obvious choices, they’re sometimes excluded. The reduced list 2, 5, 10, 25 is then called the **proper factors**.

We’ve produced a table with multiples of every number up to 100, to be used as a quick reference.

Another word that you might see is **divisor**. This is because the factors, or divisors, of a number are the only integers that divide it exactly without leaving a remainder or fraction, e.g.:

50 ÷ 10 = 5

50 ÷ 5 = 10

4 is an example of a number that is *not* a factor of 50, and you can see that

^{1}⁄

_{2}

or alternatively,

### Prime Factors of 50

A number is **prime **if only 1 and itself will multiply into it. You could also say that a prime number has *no proper factors.*

It should come as no surprise then that a **prime factor** is a factor of a number that is also prime.

As for 50, the only factors that are prime are 2 and 5.

## Factorizing 50

There are a number of different ways to factorize a number.

We’ll start by finding the prime factorization of 50, which is simpler and faster than finding all the factors directly, then the prime factorization can be used to find them *all*.

### The Prime Factorization of 50

We’ve already established that 50’s prime factors are 2 and 5. It turns out that 50 can be written as a product using only these numbers.

Let’s look at how we find that out now.

- Write down the number you’re breaking down.
- Split it up into a pair of numbers that divide it. If you don’t know any, try some small primes. If they divide your number exactly, they’re factors!
- Write each of the new numbers in the pair below the first number, connecting them with ‘branches’
- Circle any if they are prime.
- Repeat the process – split up any non-circled numbers into multiple pair and circle any primes. Keep going until all the numbers left at the ends of branches have been circled.

What you’re left with, if you started with 50, should look something like this:

The prime factorization of 50 is then the product of all the circled primes:

= 2 × 5

^{2}

Of course, your factor tree might not look exactly like the one above, as you’re free to split up the numbers into whichever factor pairs you like – the prime factorization will always be the same.

### How to Find the Factors of 50

#### Method 1: Using Divisibility Rules

Divisibility rules are so useful 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 get and use them for yourself!

The easiest way to find them all, is to start with number one and work out whether 50 is divisible by it.

All numbers are divisible by 1, so the answer is yes!

So 50 and 1 both divide 50 with no remainder. Now we need to check for divisibility between 1 and 50, so figure out whether 50 is divisible by 2.

So 2 and 25 both divide 50 and are factors! Next move on to 3.

This is not a whole number so 3 does not divide 50. Next work out if 4 is a factor.

Again, not a factor. So, move to 5.

Both 5 and 10 divide 50. Now we need to check the numbers between 5 and 10.

6 does not divide 50.

So 7 is not a factor either.

Now we have all the factors of 50, which are:

#### Method Two: Using Prime Factorization

Another way to find them all is to use the prime factorization.

Once you’ve got the prime factorization of a number, to find all the factors you just need to work out all of the possible combinations of primes.

Because the factors of a number divide it exactly, the prime decomposition of that number will only contain prime factors of the original number.

Also, the prime’s exponent in the prime decomposition of a factor will be less than or equal to the power of that prime in the original number.

50’s prime factorization is:

^{1}× 5

^{2}

Since there are only two different primes that divide 50, 2, and 5, you can use a multiplication table to lay things out.

To form the table: along the top, put all the powers of one prime, starting from zero and up to the power that appears in the prime factorization, e.g. with 50, for 5 you’ll need 5^{0}, 5^{1}, and 5^{2} 5^{0}, 5^{1}, and 5^{2}; then do the same for the other prime down the side: the other prime factor of 50 is 2, so that’ll be 2^{0} and 2^{1}:

Then, just fill out the table:

Look at that! The numbers in the table are exactly the factors of 50 that were found in the first section.

Have a go at using a multiplication table to find all the factors of these numbers. Find the prime factorizations first – but you’ve been given the prime factors as a hint, to start you off.

a) 56

Hint: Primes that go into 56 are 2 and 7

b) 225

Hint: Primes that go into 225 are 3 and 5.

c) 108

Hint: Primes that go into 108 are 2 and 3

a) **56 = 2 ^{3} × 7**

b) **225 = 3 ^{2} × 5^{2}**

c) **108 = 2 ^{2} × 3^{3}**

## Isn’t 50 Interesting?

50 is mathematically special in its own right, but we’ll get to that later. First, here are some other facts about 50:

- There are 50 states of America, hence the 50 stars on the American flag.
- 50% is equivalent to one half. That’s why people say “fifty-fifty” to refer to splitting into equal halves.

Here are some more mathematical facts about the number 50:

- 50 is the smallest number that can be written as the sum of two non-zero square numbers in two different ways:

50 = 1 + 49 = 1^{2}+ 7^{2}50 = 25 + 25 = 5

^{2}+ 5^{2} - 50 can also be written as the sum of 3 or 4 squares:

50 = 3^{2}+ 4^{2}+ 5^{2}50 = 1

^{2}+ 2^{2}+ 3^{2}+ 6^{2}

## To Sum Up (Pun Intended!)

We started off by finding the **prime factorization** of 50 using a **factor tree**. The result was:

^{2}

From this, since 50 only has two different **prime factors**, we could use a multiplication table to find all possible factors of 50, and the following list of factors were found:

If you’re not comfortable with how these methods worked, have a go at the questions at the end of Factorizing 50. If you’re still not sure, leave us a comment below.

Also, feel free to ask a question about any of the material discussed here, or if you have an interesting fact about the number 50 that we didn’t mention, then share it with us!

See some our other lessons on factoring:

Factors of 36

Factors of 48