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GCSE · EDEXCEL

4.3D Prime Factors, HCF and LCM: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 70 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Prime Factors, HCF and LCM: Fluency and Exam Drill

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This drill pack builds fluency in writing numbers as a product of their prime factors, and in using prime factorisation to find the highest common factor (HCF) and lowest common multiple (LCM) of two or more numbers. It then moves into exam-style problem-solving questions, including reasoning questions that ask you to explain or show your answer. This pack complements the main Topic 4.3 practice pack and does not share any questions with it. Show all of your working - method marks are available even when a final answer is wrong. Calculators are not allowed for this pack.

Write 18 as a product of its prime factors.

(Total for Question 1 is 2 marks)

Write 84 as a product of its prime factors, giving your answer in index form.

(Total for Question 2 is 2 marks)

Express 200 as a product of its prime factors, giving your answer in index form.

(Total for Question 3 is 2 marks)

Find the highest common factor (HCF) of 12 and 18.

(Total for Question 4 is 2 marks)

Find the HCF of 24 and 36.

(Total for Question 5 is 2 marks)

Find the lowest common multiple (LCM) of 4 and 6.

(Total for Question 6 is 2 marks)

Find the LCM of 8 and 12.

(Total for Question 7 is 2 marks)

Find the LCM of 9 and 15.

(Total for Question 8 is 2 marks)

a = 2³ x 5 and b = 2² x 5^2.

(a)Find the HCF of a and b.(2)

(b)Find the LCM of a and b.(2)

(Total for Question 9 is 4 marks)

Using prime factorisation, find the HCF and LCM of 30 and 45.

(a)Find the HCF of 30 and 45.(2)

(b)Find the LCM of 30 and 45.(2)

(Total for Question 10 is 4 marks)

Find the HCF of 45, 60 and 75.

(Total for Question 11 is 2 marks)

Find the LCM of 6, 8 and 10.

(Total for Question 12 is 2 marks)

Tom read that 84 out of 126 students in his year group play a musical instrument. By finding the HCF of 84 and 126, write the fraction 84/126 in its simplest form.

(Total for Question 13 is 3 marks)

Write 2³ x 3² x 5 as an ordinary number.

(Total for Question 14 is 2 marks)

Three warning lights on a factory control panel all flash together at the start of a shift. The red light then flashes every 8 seconds, the amber light every 12 seconds and the green light every 20 seconds. Find how many seconds it takes until all three lights next flash together.

(Total for Question 15 is 3 marks)

Priya is making identical party bags for her birthday. She has 60 chocolate coins and 84 jelly sweets. She wants to put the sweets and coins into the greatest possible number of bags so that every bag has the same number of chocolate coins and the same number of jelly sweets, with none left over.

(a)Find the greatest number of party bags Priya can make.(2)

(b)Work out how many chocolate coins and how many jelly sweets will be in each bag.(2)

(Total for Question 16 is 4 marks)

The Venn diagram below shows the prime factors of 72 and 90. In the region for 72 only, the numbers 2 and 2 are listed. In the overlapping region, the numbers 2, 3 and 3 are listed. In the region for 90 only, the number 5 is listed.

Figure (to be drawn): Two-circle Venn diagram labelled 72 and 90; the layout is fully described in words within the question text above.

(a)Use the Venn diagram to write down the HCF of 72 and 90.(2)

(b)Use the Venn diagram to write down the LCM of 72 and 90.(2)

(Total for Question 17 is 4 marks)

Show that HCF(48, 180) x LCM(48, 180) = 48 x 180.

(Total for Question 18 is 3 marks)

At a bus station, three routes all depart together at 07:00. Route A departs every 15 minutes, Route B every 20 minutes and Route C every 24 minutes.

(a)Find the next time after 07:00 that all three routes depart together.(2)

(b)The bus station is open from 07:00 until 19:00 on the same day. Work out how many more times, after 07:00, all three routes will depart together before the station closes.(2)

(Total for Question 19 is 4 marks)

n = 2³ x 3² x 5. State, with a reason, whether each of the following is a factor of n: (i) 6 (ii) 20 (iii) 25

(Total for Question 20 is 3 marks)

Amir is tiling a rectangular hallway floor that measures 210 cm by 252 cm. He wants to use the largest possible square tiles so that the floor is covered exactly, with no tiles cut.

(a)Find the side length of the largest square tile that can be used.(2)

(b)Work out how many of these tiles are needed to tile the floor exactly.(2)

(Total for Question 21 is 4 marks)

a = 2^p x 3⁴ and b = 2³ x 3^q, where p and q are positive integers. Given that HCF(a, b) = 2² x 3² and LCM(a, b) = 2³ x 3⁴, find the value of p and the value of q.

(Total for Question 22 is 3 marks)

Find the smallest positive integer n such that n is a multiple of both 18 and 24, and n is also a perfect square.

(Total for Question 23 is 4 marks)

Mark scheme · 4.3D Prime Factors, HCF and LCM: Fluency and Exam Drill

Question 1

M1 correct first split of 18 shown, e.g. 2 x 9, or a factor tree with at least one prime factor correct
A1 2 x 3² oe, fully correct in index form
Answer: 2 x 3²

Question 2

M1 at least two correct prime factors identified, e.g. 2, 2, 3, 7 from a factor tree or repeated division
A1 2² x 3 x 7 oe, fully correct in index form
Answer: 2² x 3 x 7

Question 3

M1 at least two correct prime factors identified, e.g. 2, 2, 2, 5, 5
A1 2³ x 5² oe, fully correct in index form
Answer: 2³ x 5²

Question 4

M1 12 = 2² x 3 and 18 = 2 x 3², or an equivalent listing/comparison of factors
A1 6 cao
Answer: 6

Question 5

M1 24 = 2³ x 3 and 36 = 2² x 3², or an equivalent listing/comparison of factors
A1 12 cao
Answer: 12

Question 6

M1 4 = 2² and 6 = 2 x 3, or a correct list of multiples of each number
A1 12 cao
Answer: 12

Question 7

M1 8 = 2³ and 12 = 2² x 3, or a correct list of multiples of each number
A1 24 cao
Answer: 24

Question 8

M1 9 = 3² and 15 = 3 x 5, or a correct list of multiples of each number
A1 45 cao
Answer: 45

Question 9

(a) M1 correct lowest power of each common prime identified, 2² and 5¹
(a) A1 20 cao
(a) Answer: 20
(b) M1 correct highest power of each prime identified, 2³ and 5²
(b) A1 200 cao
(b) Answer: 200

Question 10

(a) M1 30 = 2 x 3 x 5 and 45 = 3² x 5 shown
(a) A1 15 cao
(a) Answer: 15
(b) M1 highest power of each prime identified, 2¹, 3² and 5¹
(b) A1 90 cao
(b) Answer: 90

Question 11

M1 45 = 3² x 5, 60 = 2² x 3 x 5 and 75 = 3 x 5² all shown
A1 15 cao
Answer: 15

Question 12

M1 6 = 2 x 3, 8 = 2³ and 10 = 2 x 5 all shown
A1 120 cao
Answer: 120

Question 13

M1 84 = 2² x 3 x 7 and 126 = 2 x 3² x 7, leading to HCF = 42
M1 84 and 126 both divided by 42 (or their stated HCF)
A1 2/3 oe, fully simplified
Answer: 2/3

Question 14

M1 2³ = 8 and 3² = 9 (or 8 x 9 = 72) evaluated correctly
A1 360 cao
Answer: 360

Question 15

M1 8 = 2³, 12 = 2² x 3 and 20 = 2² x 5 all shown (or an equivalent listing of multiples of all three numbers)
M1 highest power of each prime identified, 2³, 3¹ and 5¹
A1 120 cao (accept 2 minutes)
Answer: 120 seconds (2 minutes)

Question 16

(a) M1 60 = 2² x 3 x 5 and 84 = 2² x 3 x 7 shown, or an equivalent factor comparison
(a) A1 12 cao
(a) Answer: 12
(b) M1 60 divided by their answer to part (a), or 84 divided by their answer to part (a) (ft)
(b) A1 5 chocolate coins and 7 jelly sweets, both stated
(b) Answer: 5 chocolate coins and 7 jelly sweets per bag

Question 17

(a) M1 the numbers in the overlapping region multiplied together, 2 x 3 x 3
(a) A1 18 cao
(a) Answer: 18
(b) M1 all of the numbers in the whole diagram multiplied together, 2 x 2 x 2 x 3 x 3 x 5
(b) A1 360 cao
(b) Answer: 360

Question 18

M1 48 = 2⁴ x 3 and 180 = 2² x 3² x 5 used to find HCF = 12
M1 48 = 2⁴ x 3 and 180 = 2² x 3² x 5 used to find LCM = 720
A1 12 x 720 = 8640 and 48 x 180 = 8640 both evaluated and shown to be equal, with a concluding statement
Answer: 12 x 720 = 8640 = 48 x 180, so the statement is true

Question 19

(a) M1 15 = 3 x 5, 20 = 2² x 5 and 24 = 2³ x 3 used to find LCM = 120 (minutes), or an equivalent multiples method
(a) A1 09:00 cao
(a) Answer: 09:00
(b) M1 07:00 to 19:00 identified as 12 hours (720 minutes), divided by their LCM from part (a) (ft)
(b) A1 6 cao
(b) Answer: 6

Question 20

B1 (i) yes, with reason that n contains both 2¹ and 3¹ (since n has 2³ and 3²)
B1 (ii) yes, with reason that n contains both 2² and 5¹ (since n has 2³ and 5¹)
B1 (iii) no, with reason that n only contains 5¹, but 25 needs 5²
Answer: (i) yes (ii) yes (iii) no

Question 21

(a) M1 210 = 2 x 3 x 5 x 7 and 252 = 2² x 3² x 7 used to find the HCF
(a) A1 42 cm cao
(a) Answer: 42 cm
(b) M1 210 and 252 each divided by their answer to part (a) (ft), and the two results multiplied together
(b) A1 30 cao
(b) Answer: 30

Question 22

M1 the power of 2 in the HCF used correctly: min(p, 3) = 2, leading to p = 2
M1 the power of 3 in the HCF (or LCM) used correctly: min(4, q) = 2, leading to q = 2
A1 p = 2 and q = 2 both stated, with the LCM powers checked as consistent
Answer: p = 2, q = 2

Question 23

M1 18 = 2 x 3² and 24 = 2³ x 3 used to find LCM(18,24) = 2³ x 3² = 72
M1 recognising that for n to be a perfect square, every prime in its factorisation must have an even power
M1 identifying that the power of 2 (currently 2³) is the only odd power, so 72 must be multiplied by a further 2
A1 144 cao
Answer: 144

Prime Factors, HCF and LCM: Fluency and Exam Drill - Worksheets, Questions and Revision