# Find Factors

In this worksheet, students will practise finding factors of numbers with up to two digits to aid application and identification of these special numbers in problems.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Number, Number Operations and Integers

Curriculum subtopic:   Structure and Calculation, Whole Number Theory

Difficulty level:

### QUESTION 1 of 10

In number theory, there are lots of different types of numbers: odds, evens, primes, squares, etc.

An understanding of what makes these numbers special and knowledge of which numbers fall into these categories can help you solve complex problems quickly and with confidence.

In this activity, we're going to look exclusively at factors.

A factor is a number which you can divide by that doesn't leave a remainder.

For example, 4 is a factor of 20 because if you divide 20 by 4, you get a whole number (5).

e.g. Find all the factors of 18.

There's a great little trick that will help you find all the factors - they always come in pairs which multiply to your target number!

Ask: "What do I multiply 1 by to get 18?"

The answer, 18, is also a factor.

1, 18

Next, step up from 1.

Is 2 a factor?

Yes.

What does it go with?

9 (2 x 9 = 18)

1, 18

2, 9

Step it up again.

Is 3 a factor?

Yes.

What does it go with?

6 (3 x 6 = 18)

1, 18

2, 9

3, 6

Is 4 a factor?

No.

Is 5 a factor?

No

So why can I stop now?

The next number I would look at is 6 which I already know is a factor.

My numbers on the left go up, the numbers on the right come down and they eventually meet in the middle.

Job done!

So the factors of 18 are: 1, 2, 3, 6, 9, 18.

e.g. Find all the factors of 36.

If we follow the same idea as before, starting at 1 and working our way up, we should get the following factor pairs.

1 x 36

2 x 18

3 x 12

4 x 9

6 x 6

So the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Be careful not to write 6 twice, as it can only be a factor once!

In this activity, you will apply the process described above to find factors of numbers with up to two digits.

Complete the sentence below to summarise how to identify factors.

Complete the following sentence.

For each of the options below, select if the list of factors is accurate and complete or not.

Find all the factors of 40.

Put your answers in ascending (smallest to biggest) order with 1 space between them.

You may want to write your list on scrap paper first.

No other punctuation is necessary.

Which of the following lists is the correct complete list of factors of 64?

1, 2, 4, 8, 16, 32, 64

1, 2, 4, 8, 8, 16, 32, 64

1, 2, 4, 6, 8, 16, 32, 64

Which of the following numbers are factors of 320?

1

3

40

6

8

9

15

10

For each of the options below, match them with the total number of factors each has.

E.g. 12 has 6 factors: 1, 2, 3, 4, 6, 12.

## Column B

25
9
100
3
37
8
40
2

Find all the factors of 28.

Put your answers in ascending (smallest to biggest) order with 1 space between them.

You may want to write your list on scrap paper first.

No other punctuation is necessary.

Does the number 4 or 49 have the most factors?

4

49

They have the same

Complete the sentence below to describe the relationship between prime numbers and factors

4

49

They have the same

• Question 1

Complete the sentence below to summarise how to identify factors.

EDDIE SAYS
A quick definition check to start here! The key fact to remember is that when you divide by a factor, you do not have any remainder. Another way to think about factors is the times tables in which the target number will appear as an answer.
• Question 2

Complete the following sentence.

EDDIE SAYS
In the Intro, we talked about the fact that factors always come in pairs. So how can a number have an odd number of factors? It has to be a square number. e.g. The factors of 16 are: 1, 2, 4, 8, 16. We only write the 4 once, even though 4 x 4 = 16. Remember that only square numbers will have an odd number of factors - this can help you identify if you are missing any factors.
• Question 3

For each of the options below, select if the list of factors is accurate and complete or not.

EDDIE SAYS
It's really easy to miss out one factor in a list. The easiest way to check this for each number in the list of factors is to ask: "Is the number I multiplied by also in my list?" For 15, we have a factor of 3. This should go with 5 (3 x 5 = 15) but 5 isn't in the list. Therefore, this is an incomplete list of factors. For 12 and 24, the list of factors are accurate and complete. We can check this by ensuring we have an even number too, as neither 12 and 24 are square numbers.
• Question 4

Find all the factors of 40.

Put your answers in ascending (smallest to biggest) order with 1 space between them.

You may want to write your list on scrap paper first.

No other punctuation is necessary.

1 2 4 5 8 10 20 40
EDDIE SAYS
Remember to look for pairs of factors: 1 x 40 2 x 20 4 x 10 5 x 8 Did you get all 8 factors? And type them in carefully and in the correct order? It's easy to miss a pair, so be wary of this.
• Question 5

Which of the following lists is the correct complete list of factors of 64?

1, 2, 4, 8, 16, 32, 64
EDDIE SAYS
64 has 7 factors. Did you find your pairs of calculations? 1 x 64 2 x 32 4 x 16 8 x 8 It has an odd number of factors as it is a square number. The most common mistake is putting the repeated factor in for a square number. 8 x 8 = 64, but we only need to put the 8 in our list once! There's also a list that has an incorrect factor in. Did you spot that?
• Question 6

Which of the following numbers are factors of 320?

1
40
8
10
EDDIE SAYS
Whenever you are faced with a question like this with options, try to divide the target number by each option. e.g. Do you get a whole number if you calculate 320 ÷ 15? No, so 15 isn't a factor then. How did you get on with those divisions?
• Question 7

For each of the options below, match them with the total number of factors each has.

E.g. 12 has 6 factors: 1, 2, 3, 4, 6, 12.

## Column B

25
3
100
9
37
2
40
8
EDDIE SAYS
25 and 100 are square numbers so they must have an odd number of factors. To work out the number of factors of each, you can write the calculations. E.g. 40 > 1 x 40; 2 x 20; 4 x 10; 8 x 5 37 only has two factors because it's a prime number. This means that it can only be divided by itself and 1.
• Question 8

Find all the factors of 28.

Put your answers in ascending (smallest to biggest) order with 1 space between them.

You may want to write your list on scrap paper first.

No other punctuation is necessary.

1 2 4 7 14 28
EDDIE SAYS
Remember, you're looking for pairs of factors, unless you're dealing with a square number. Here our pairs are: 1 x 28 2 x 14 4 x 7 Did you get all 6? And in the correct order too? It's easy to miss a pair so be sure to take your time and write them all out.
• Question 9

Does the number 4 or 49 have the most factors?

They have the same
EDDIE SAYS
A common misconception is that the bigger the number is, the more factors it will have. This isn't true. Both of these numbers have 3 factors: 4 = 1, 2, 4 49 = 1, 7, 49 They both have an odd number of factors as they are both square numbers: 2 x 2 = 4 7 x 7 = 49
• Question 10

Complete the sentence below to describe the relationship between prime numbers and factors

EDDIE SAYS
A prime number must have exactly two factors. Itself and 1. A composite number is any number with more than two factors. 1 is the only number which is neither prime or composite, as it has only one factor. Great focus, that’s another activity completed! Why not revise prime numbers or division if you found this challenging?
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