KS3 Maths · Year 7 · Chapter 1: Working with numbers · Lesson 2

Factors and highest common factor

Year 7About 60 minutesNo calculator49 marks

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Date

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Quick questions to start.

A1
Needed today
Work out $126 \div 9$ .
[1 mark]
Answer
A2
Needed today
Write $\frac{10}{25}$ in its simplest form.
[1 mark]
Answer
A3
Last lesson
Work out $- 6 \times - 7 \div - 3$ .
[1 mark]
Answer
A4
Last lesson
Work out $(- 2)^{3} \times 5$ .
[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

Write down all the factors of $48$ .

Hint Find the factor pairs:

1 \times 48

2 \times 24

, and so on. Stop when the pairs start to repeat.

B1
Write down all the factors of $84$ .
[2 marks]
Answer

Example 2

Find the highest common factor (HCF) of $42$ and $70$ .

Hint List the factors of each number, then find the largest number in both lists.

B2
Write down all the common factors of $60$ and $84$ . Then write down the HCF of $60$ and $84$ .
[3 marks]
Answer $\dots\dots\dots\dots$ ; HCF $\dots$

Example 3

Use the HCF to write $\frac{56}{84}$ in its simplest form.

Hint Find the HCF of

56

and

84

. Divide the numerator and the denominator by it.

B3
Use the HCF to write $\frac{52}{91}$ in its simplest form.
[2 marks]
Answer

Example 4

A choir has $45$ sopranos and $60$ altos. The choir leader puts them into groups. Every group has the same number of singers, and sopranos and altos are not mixed. What is the largest possible size of a group? How many groups are there?

Hint A group size must divide

45

and

60

exactly, so it is a common factor. The largest is the HCF.

B4
Raj has $72$ red beads and $90$ blue beads. He puts all of them into bags. Every bag holds the same number of red beads and the same number of blue beads. What is the largest number of bags he can fill? How many red beads and how many blue beads go in each bag?
[3 marks]
Answerbags $\dots$ red $\dots$ blue $\dots$

Example 5

A floor is $3.6$ m long and $2.7$ m wide. It is covered with square tiles, all the same size, with no tiles cut. What is the side length of the largest tile that can be used? How many of these tiles are needed?

Not drawn accurately

Hint Change the lengths to centimetres so they are whole numbers. The tile side must divide both lengths.

B5
A sheet of card measures $84$ cm by $60$ cm. It is cut into squares, all the same size, as large as possible, with no card left over. What is the side length of each square? How many squares are there?
[3 marks]
Not drawn accurately
Answerside $\dots\dots$ cm $\dots\dots$ squares

Practice

Year 7

C1
Write down all the common factors of $36$ and $90$ .
[2 marks]
Answer
C2
Find the HCF of $39$ , $65$ and $91$ .
[2 marks]
Answer

C3
Which whole number from $30$ to $40$ has the most factors? Write down the number and how many factors it has.
[2 marks]
Answernumber $\dots\dots$ factors $\dots\dots$
C4
Find the largest whole number that leaves a remainder of $2$ when it is divided into $50$ and when it is divided into $74$ .
[2 marks]
Answer
C5
The HCF of $24$ and a number $n$ is $8$ . $n$ is between $60$ and $70$ . Find $n$ .
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
This question is about factors of $40$ .
(a)
Write down all the factors of $40$ .
[2 marks]
Answer
(b)
Find the highest common factor (HCF) of $40$ and $64$ .
[1 mark]
Answer
Total for D1: 3 marks

D2
Sana says, "The HCF of $18$ and $54$ is $9$ , because $9$ is a factor of both numbers."
Sana is wrong. Explain why, and write down the correct HCF.
[2 marks]
D3
A florist has $84$ roses and $126$ tulips. She uses all of them to make bunches. Every bunch is the same: the same number of roses and the same number of tulips.
(a)
Work out the greatest number of bunches she can make.
[2 marks]
Answer
(b)
How many roses and how many tulips are in each bunch?
[2 marks]
Answer $\dots\dots$ roses and $\dots\dots$ tulips
(c)
The next day she has $84$ roses, $126$ tulips and $35$ lilies. She makes bunches in the same way, using all three kinds of flower. Work out the greatest number of bunches she can make.
[2 marks]
Answer
Total for D3: 6 marks
D4
A rectangular hall floor is $5.4$ m long and $4.2$ m wide. It is covered with square tiles, all the same size, with no tiles cut. The tiles are as large as possible.
Not drawn accurately
(a)
Work out the side length of each tile, in centimetres. You must show your working.
[3 marks]
Answer
(b)
How many tiles are needed?
[1 mark]
Answer
Total for D4: 4 marks
D5
Two ribbons are $96$ cm and $156$ cm long. They are cut into pieces that are all the same length, as long as possible, with no ribbon left over. How many pieces are there altogether? You must show your working.
[3 marks]
Answer

Extension

Stretch

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

E1
The HCF of two numbers is $18$ . The larger number is $54$ more than the smaller number. Both numbers are less than $100$ . Find every possible pair of numbers.
[4 marks]
Hint 1 · What to try
Both numbers are multiples of $18$ . List the multiples of $18$ below $100$ .
Hint 2 · The first line
The multiples are $18$ , $36$ , $54$ , $72$ , $90$ . Pick pairs that differ by $54$ .
Hint 3 · The full method
$18$ and $72$ , and $36$ and $90$ , differ by $54$ . Check each HCF is $18$ , not more: both are $18$ .
Answer $\dots\dots$ and $\dots\dots$ $\dots\dots$ and $\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can list all the factors of a number.
I can find the highest common factor of two or three numbers.
I can use the HCF to simplify fractions and solve sharing problems.

Answers and mark scheme

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