According to Aristotle, 56 is the number of layers of the Universe. He held that Earth is the first layer, with 55 crystalline spheres above it.

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This lesson will walk you through all 56’s factors, how to find the factors and other exciting factor facts!

If you want a quick reference for factors, this factors and multiples table will be more useful.

A factor is a number that goes into another number perfectly. For example, 2 is a factor of 10 because it goes into 10 perfectly without leaving a remainder.

If you multiply 2 by 5, you get 10, which means both 2 and 5 are factors of 10.

You will find out what 56’s factors are, how to find factor pairs and all about prime factorization in this lesson.

Once you understand this, you will be able to find the factors of any number!

Contents

56’s Factors Pairs Primes Factorizing 56 How to Find 56’s Factors Prime Factorization Isn’t 56 Interesting? To Sum Up (Pun Intended!)

## All the Factors of 56

The factors of 56 are **1, 2, 4, 7, 8, 14, 28, and 56**.

56 can be divided by itself and one, but it is also divisible by other numbers, so it is a **composite number**, and there are more than two factors!

As you have just learned, a **factor** is one number that divides another without leaving a remainder.

You can write a number in terms of its factor pairs or a product of its prime factors.

### Factor Pairs of 56

The simplest way to find all the factors of a number is to work out the factor pairs.

Factor pairs are sets of two numbers that multiply together to make another number.

So you need to find out which two numbers, ** X** and

**multiply together to make 56. This is written as:**

*Y*(

*X*,

*Y*)

Factor Pairs of 56

(1, 56) (2, 28) (4, 14) (7, 8)

Sometimes you can include negative numbers as a factor because when two negative numbers are multiplied together, they make a positive number!

It is important to remember that both numbers need to be negative to make a positive number. If one of them is negative, and one of them is positive, they will be negative.

Negative Factor Pairs(-1, -56) (-2, -28) (-4, -14) (-7, -8)

You have probably noticed that the negative factors are the same as the positive ones, only they’re negative!

However, only the positive ones are being referred to unless said otherwise when talking about the factor pairs.

### Is 56 Abundant?

If the sum of the positive factors is bigger than twice the number itself, it is known as an **abundant number**.

The sum of the positive factors is:

120 = 1+2+4+7+8+14+28+56

Double 56 is 112, and the sum of all its factors is 120. This makes 56 an abundant number: the sum of its factors is greater than double itself.

### Prime Factors of 56

A prime number is one that cannot be divided by anything other than 1 or itself. For example, 3.

3 ÷ 1 = 3 3 ÷ 3 = 1

3 cannot be divided by anything else so it is a prime number.

56 has only two prime factors, which are 2 and 7.

You can use these prime factors to write 56 as a product of its prime factors.

56=2 × 2 × 2 × 7

You can work this out by using any two factor pairs and keep dividing them down until you have prime numbers.

This is a **factor tree**. It starts with the original number at the top, which is broken down into factor pairs.

When a prime is found, it is circled and the other number is broken down further.

The prime number, 2, appears on three occasions, so this can be written more simply using power or exponent form.

56 = 23 × 7

The number 56 also has other prime numbers. You can express 56 as the sum of consecutive primes.

56 = 3 + 5 + 7 + 11 + 13 + 17

Why Find the Prime Factors?

Finding all the prime factors of a number can be useful in finding the **Greatest Common Factor** (**GCF**) of two numbers, as well as the **Least Common Multiple** (**LCM**) of two numbers.

Prime factorization is also useful in the real world in cryptography. It can take an exceptionally long time to find a large number’s prime factorization, even for computers!

## Factorizing 56

### How to Find the Factors of 56

To find all the factors, you need to start with 1 and work through each number one by one to 56, checking if there is a remainder when 56 is divided by each number. If there is no remainder, the number is a factor.

Each time you find a factor pair, you reduce the interval of factors you are looking for to numbers within such factor pair intervals.

We will share a few tricks with you to make this a quick job.

The ProcessStarting at 1…

56 ÷ 1 = 56

There is no remainder, so you have confirmed 1 and 56 are a factor pair. You now need to find factors between 1 and 56.

56 ÷ 2 = 28

So, 2 and 28 are a factor pair. You now need to find factors between 2 and 28.

56 ÷ 3 = 18.67

So, 3 is not a factor. Continue with this process with 4, 5, and 6.

You will see there are no factors up to 6.

56 ÷ 7 = 56

7 and 8 are a factor pair. There are no integers between 7 and 8, so you have found all the factor pairs of 56.

So, in total you have the following factors:

1, 2, 7, 8, 14, 28, 56

Here are a few tricks for spotting factors:

All integers have a factor of 1Any even number has a factor of 2If the sum of the digits of a number is a multiple of 3 then 3 is a factorNumbers with a factor of 4 must be even and if the last two digits are 00 or divisible by 4 then 4 is a factorAny number with the last digit 0 or 5 has a factor of 5Any number that is even and the sum of its digits are a multiple of 3 will have a factor of 6There isn’t a simple trick for spotting factors of 7!If the last 3 digits of a number are divisible by 8 it will have a factor of 8If the sum of a number’s digits is a multiple of 9 then it has a factor of 9Numbers with the last digit 0 will have a factor of 10These are so useful in fact, that we made a printable table of the divisibility rules from 2 to 15, plus a worksheet to practice, for you to keep with you–for your own use. Just click or tap the images below!

Now, some practice.

Using the method above, find the factors of: a) 24 b) 37 c) 72 (Open to show solutions)a) 1, 2, 3, 4, 6, 8, 12, 24

b) 37 is a prime number, so the only factors are 1, 37

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

### Prime Factorization

Prime factorization is finding all the prime numbers which multiply together to make the original number.

The easiest method to find all the prime numbers is to use a prime factor tree.

At the start of the tree, you have 56.

Begin by choosing two factor pairs that multiply together to make the number.

Choose the factor pair with the smallest prime number in it, e.g., 2 and 28, and add them as branches to the tree.

As 2 is a prime number, you can circle it to show it is the final factor on this branch, and it can’t be divided anymore.

Now you need to continue breaking down 28 into its factors by finding the smallest prime number that divides 28.

It is divisible by 2 again, so…

Again 2 can be circled. You still need to break down 14, which, as it turns out, is divisible by 2 again!

Both 2 and 7 are prime numbers, so you can circle both.

Just like the factor tree you were shown before, the end of every branch is circled, so you have completed your own prime factor tree!

You can now express 56 as a product of the circled prime factors.

56 = 2 × 2 × 2 × 7

This can be neatened up by using the power or exponent form:

56 = 23 × 7

Write each number as a product of its prime factors. a) 48 b) 90 c) 104 (Open to show solutions)

a) 2×2×2×2×3=24×3

b) 2×3×3×5=2×32×5

c) 2×2×2×13=23×13

## Isn’t 56 Interesting?

There are several ways of expressing 56, other than its factors, as it has some interesting properties.

56 is the sum of the first 6 triangular numbers.

56 = 1+3+6+10+15+21

A triangular number counts objects arranged in an equilateral triangle.

This makes 56 the 6th tetrahedral number as it is the sum of the first 6 triangular numbers, making it a triangular pyramidal number.

You can also express 56 as the sum of consecutive even or odd numbers.

For even numbers…

56 = 2+4+6+8+10+12+14

For odd numbers…

56 = 11+13+15+17

and

56 = 27+29

56 is also part of a Pythagorean triple. Pythagorean triples are sets of 3 numbers which satisfy:

a2 + b2 = c2

In this case…

332 + 562 = 652

This represents a right-angle triangle with lengths 33, 56 and 65, where the longest side, or the **hypotenuse** is 65.

### Did You Know?

56 is the atomic number of the element Barium.

Pythagoreans associated a polygon of 56 sides with Typhon, a Greek mythical creature described as a monstrous serpentine giant.

It is impossible to construct a perfect regular 56-sided polygon without a computer’s aid; however, a close approximation was discovered at Stonehenge called Aubrey Holes.

In the year 56AD, the war between Rome and Parthia started due to the invasion of Armenia.

And strangely…There is a city in Arkansas called Fifty-Six!

## To Sum Up (Pun Intended!)

Factors divide exactly into a number without leaving a remainder. You can express the factors of 56 in terms of its factor pairs or its prime factors.

The factor pairs of 56 are given in the form:

(

*X*,

*Y*)

This is because ** X** and

**are two numbers that multiply together to make 56.**

*Y*To find all the factor pairs, you need to check each pair of numbers from 1 to 56, which multiply together to make 56. Visit this link for our printable Divisibility Rules.

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You found that the factors of 56 are **1, 2, 4, 7, 8, 14, 28 and 56**.

Please leave a comment below if you have any questions and let us know how you did with the challenges!