KS3 Maths · Year 7 · Chapter 1: Working with numbers · Lesson 5

Prime factors

Year 7About 60 minutesNo calculator49 marks

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Quick questions to start.

A1
Needed today
Write down all the prime numbers between $20$ and $40$ .
[1 mark]
Answer
A2
Needed today
Work out $2^{3} \times 5$ .
[1 mark]
Answer
A3
Last lesson
Work out $(- 4)^{3}$ .
[1 mark]
Answer
A4
From chapter 1
Find the HCF of $36$ and $60$ .
[1 mark]
Answer

Example, then your turn

Year 7

Your teacher works through each example with you. Then try the one beside it.

Example 1

Use a factor tree to write $126$ as a product of its prime factors. Give your answer in index form.

Hint Split the number into two factors. Keep splitting until every branch ends in a prime. Ring the primes.

B1
Use a factor tree to write $132$ as a product of its prime factors. Give your answer in index form.
[2 marks]
Answer

Example 2

Use the division method to write $392$ as a product of its prime factors in index form.

Hint Divide by

2

as many times as you can, then try

3

, then

5

, then

7

, until you reach

1

B2
Use the division method to write $270$ as a product of its prime factors in index form.
[2 marks]
Answer

Example 3

What number is $2^{3} \times 3^{2} \times 7$ ?

Hint Work out each power first, then multiply.

B3
What number is each of these?
(a) $2^{4} \times 7$
(b) $2^{2} \times 3 \times 5^{2}$
[2 marks]
Answer(a) $\dots$ (b) $\dots$

Example 4

$168 = 2^{3} \times 3 \times 7$ and $180 = 2^{2} \times 3^{2} \times 5$ . Use a Venn diagram to find the HCF and the LCM of $168$ and $180$ .

Hint Put the primes the two numbers share in the middle. The HCF is the product of the middle. The LCM is the product of every number in the diagram.

B4
$154 = 2 \times 7 \times 11$ and $220 = 2^{2} \times 5 \times 11$ . Find the HCF and the LCM of $154$ and $220$ .
[3 marks]
AnswerHCF $\dots$ LCM $\dots$

Example 5

Find the smallest whole number you can multiply $98$ by to get a square number.

Hint In a square number, every prime appears an even number of times.

B5
Find the smallest whole number you can multiply $108$ by to get a square number.
[2 marks]
Answer

Practice

Year 7

C1
Write $2000$ as a product of its prime factors. Give your answer in index form.
[2 marks]
Answer
C2
$2^{a} \times 5^{b} = 400$ . Find the values of $a$ and $b$ .
[2 marks]
Answer $a = \dots$ $b = \dots$

C3
$540 = 2^{2} \times 3^{3} \times 5$ . Write $1080$ as a product of its prime factors in index form.
[2 marks]
Answer
C4
Use prime factors to find the HCF of $270$ and $252$ .
[2 marks]
Answer
C5
$221$ is the product of two prime numbers. Find the two prime numbers.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
This question is about the numbers $616$ and $440$ .
(a)
Write $616$ as a product of its prime factors. Give your answer in index form.
[2 marks]
Answer
(b)
$440 = 2^{3} \times 5 \times 11$ . Find the HCF of $616$ and $440$ .
[2 marks]
Answer
(c)
Find the LCM of $616$ and $440$ .
[2 marks]
Answer
Total for D1: 6 marks

D2
Jo says, " $135$ written as a product of its prime factors is $3 \times 3 \times 15$ ."
Jo is wrong. Explain why, and write $135$ correctly as a product of its prime factors.
[2 marks]
D3
Ali and Bea run laps of a track. They start together at the start line. Ali takes $75$ seconds to run one lap. Bea takes $105$ seconds to run one lap. How long is it until they are next at the start line together? Give your answer in minutes and seconds.
[4 marks]
Answer $\dots$ minutes $\dots$ seconds
D4
$x = 2^{3} \times 3 \times 5^{2}$ and $y = 2^{2} \times 3^{2} \times 5$ .
(a)
Find the HCF of $x$ and $y$ . Give your answer in index form.
[2 marks]
Answer
(b)
Find the LCM of $x$ and $y$ . Give your answer in index form.
[2 marks]
Answer
Total for D4: 4 marks
D5
This question is about the number $1764$ .
(a)
Write $1764$ as a product of its prime factors. Give your answer in index form.
[2 marks]
Answer
(b)
Use your answer to part (a) to work out $1764$ .
[2 marks]
Answer
Total for D5: 4 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Three whole numbers next to each other, such as $4$ , $5$ and $6$ , are multiplied together. The answer is $2730$ . Find the three numbers.
[4 marks]
Hint 1 · What to try
Write $2730$ as a product of its prime factors.
Hint 2 · The first line
$2730 = 2 \times 3 \times 5 \times 7 \times 13$ . Group these primes into three numbers that are close together.
Hint 3 · The full method
One of the numbers must be $13$ . $2 \times 7 = 14$ and $3 \times 5 = 15$ , so the numbers are $13$ , $14$ and $15$ .
Answer $\dots$ , $\dots$ , $\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write a number as a product of its prime factors in index form.
I can use prime factors to find the HCF and the LCM of two numbers.
I can use prime factors to find a square root or make a square number.

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