# What is a Factor? What is a factor in Maths?

A factor in maths is one of 2 or more numbers that divide into a number without a remainder, making it a whole number. In other words, a factor is a number that divides another number evenly. There are no numbers left over after the division process.

For instance, 5 x 2 = 10, so 5 and 2 are factors of 10.

Any number can have a large number of factors. Many numbers have an even number of factors, but square numbers have an odd number.

For instance, 25 is a square number because it is the square of 5. Therefore, its factors are 1, 5, and 25.

Factors are found not just in numerical sums but also in algebraic equations.

Factoring is the opposite of expanding. You can find out more about factoring and expanding by reading on. There are also some teacher-made resources listed to support your teaching about factors. Top of Form

How to calculate factors in Maths

One of the easiest ways children calculate factors is to use a factor tree. It is a simple root-and-branch approach to determine which numbers can be multiplied to reach a particular number.

The main aim of using this method is to find prime factors, which are numbers that cannot be factored down anymore. Prime factors must be prime numbers because they are only divisible by themselves and by 1.

A factor tree for the number 24, for example, would look like this:

24

4 x 6

2 x 2 x 2 x 3

Therefore, the result of factorizing the number 24 is 2 x 2 x 2 x 3, with 2 and 3 both being prime numbers.

Every whole number that isn’t 1 can be expressed as the sum of its prime factors.

Factor pairs

What are factor pairs?

Factor pairs are a combination of two numbers that act as factors of a multiple, giving a known product or number when multiplied together.

Let’s look at the number 20. An example of factor pairs of 20 would be the multiplication sum of 10 x 2.

10 x 2 = 20

10 and 2 are the factor pair, and 20 is the product of this factor pair.

For the multiple of 20, the factors of 10 and 2 together aren’t the only factor pair that exist. For example, 5 and 4; are a factor pair of the multiple of 20, as 1 and 20.

Learning multiplication tables well can help children to understand factor pairs and identify them quickly. Why not ask your children to match all the possible factor pairs for a number to demonstrate the concept?

Different types of factor

Prime Factor

Many numbers have an even number of factors, while a prime number has only two factors; the prime number itself and the number 1. It means it only has one-factor pair.

It means a prime factor is a factor that is also a prime number.

In other words, it is a number greater than 1 but cannot be divided precisely except by itself or by 1.

Common Factor

When working out the factors of two or more numbers, you will often find that their factors overlap. We call these overlapping numbers common factors.

For example, using the example of 18 and 24, common factors which will multiply into both include 1, 2, 3, and 6.

Highest Common Factor (HCF)

As you may have guessed, the highest common factor in a sum is the highest number of the common factors you have identified.

For example, the highest common factor of 24 and 6 is 6. It is because 6 goes into 6 once, and 6 goes into 24 four times.

What are the factors of 100?

The factors of 100 are all the numbers that make 100 when two are multiplied. Similarly, factor pairs of 100 are the whole positive or negative numbers that equal 100.

Factor pairs cannot include a fraction or a decimal number.

To find factors of 100, we must first take the numbers 1 and 100 and begin discovering the other pairs of numbers, which, when multiplied together, make 100.

Factors of 100:

• 1
• 2
• 4
• 5
• 10
• 20
• 25
• 50
• 100

What are the factor pairs of 100? To find both the positive and negative factor pairs of 100, take a pair and multiply the two numbers together to get 100. Here are some examples:

Positive Factor Pairs of 100

1 × 100 = 100 ⇒ (1, 100)

2 × 50 = 100 ⇒ (2, 50)

4 × 25 = 100 ⇒ (4, 25

5 × 20 = 100 ⇒ (5, 20)

10 × 10 = 100 ⇒ (10, 10

Negative Factor Pairs of 100

-1 × -100 = 100 ⇒ (-1, -100)

2 × -50 = 100 ⇒ (-2, -50)

-4 × -25 = 100 ⇒ (-4, -25)

-5 × -20 = 100 ⇒ (-5, -20

-10 × -10 = 100 ⇒ (-10, -10)

Factors of 100 by Prime Factorisation

Prime factorization refers to expressing a composite number as the product of the prime factors. There are three key steps to prime factorization:

To get the prime factorization of 100, we must divide it by its smallest prime factor, 2: 100 ÷ 2 = 50.

Next, divide 50 by its most minor prime factor.

Repeat this process until the product is 1.

The Division Method

In the same way that the factors of 100 can be found by multiplying numbers together, it can also be found through the division method. To carry out this division method, divide 100 by numbers, starting from 1, and see whether they leave you with a whole number or leave a remainder as well.

For example:

100 ➗ 1 = 100 (The factor is 1, and the remainder is 0)

100 ➗ 2 = 50 (The factor is 2, and the remainder is 0)

100 ➗ 4 = 25 (The factor is 4, and the remainder is 0)

100 ➗ 5 = 20 (The factor is 5, and the remainder is 0)

100 ➗ 10 = 10 (The factor is 10, and the remainder is 0)

100 ➗ 20 = 5 (The factor is 20, and the remainder is 0)

100 ➗ 25 = 4 (The factor is 25, and the remainder is 0)

100 ➗ 50 = 2 (The factor is 50, and the remainder is 0)

100 ➗ 100 = 1 (The factor is 100, and the remainder is 0)

However, if you were to divide 100 by 3, you would get a remainder of 0.333. So it means that 3 is not a factor of 100.

What is a factor in algebra?

In algebraic equations, factors are expressed differently from the sums we have seen before.

Here, factoring, or factorizing, is done by finding an expression by multiplying simpler expressions together.

For example, if you are asked to factor 2x+4:

2x is 2 lots of x.

4 is 2 lots of 2.

So, to factorize the sum, we combine the two:

2x+4=2(x+2).

The opposite of factoring is expanding.

Expanding a bracket means multiplying each term by the expression outside the bracket.

You’ll see it is much like factoring, just the other way around. Knowing how to expand will help you to remember how to factor.

We can expand the answer we got above by following this process.

For example, with the expression 2(x+2), we multiply both x and 2 by the number outside the bracket. In this case, that’s the number 2. So:

2(x+2) = 2 × x + 2× 2 = 2x+4

Both expanding and factorizing make use of the skills of simplifying algebra.

What is the difference between a factor and a multiple?

Although factors and multiples are directly linked, they are two different things.

Where factors refer to the numbers that can be multiplied to reach a number, a multiple is a result after the factors are multiplied.

So, multiples result from a multiplication sum, whereas factors are the numbers that create the sum. Therefore, they are opposite concepts to each other.