Find the highest common factor (HCF) of 54 and 120
Worked Solution & Answer
54 = 2 × 3³ M1
120 = 2³ × 3 × 5. Common prime factors: 2 × 3 = 6. A1
120 = 2³ × 3 × 5. Common prime factors: 2 × 3 = 6. A1
Answer: 6
Prime factors HCF and LCM — Notes & Worked Examples Show / Hide
Explanation:
To express a number as a product of its prime factors, use a factor tree to split it until every branch ends at a prime number (2, 3, 5, 7, 11, 13…), then write the primes multiplied together in index form (e.g. 2³ × 3 × 5).
To find the HCF and LCM of two numbers, put their prime factors into a Venn diagram: multiply the primes in the intersection (overlap) to get the HCF, and multiply all primes in both circles to get the LCM.
Example 1: Write 140 as a product of its prime factors in index form.
140 = 2 × 70
70 = 2 × 35
35 = 5 × 7
2² × 5 × 7
Example 2: Given 60 = 2² × 3 × 5 and 84 = 2² × 3 × 7, find the HCF and LCM.
HCF = 2×2×3 = 12
LCM = 12×5×7
= 420
Write 60 as a product of its prime factors.
Worked Solution & Answer
60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 (or 2² × 3 × 5) M1 A1
Answer: 2 × 2 × 3 × 5
Write 36 as a product of its prime factors.
Worked Solution & Answer
36 = 4 × 9 = 2 × 2 × 3 × 3 = 2² × 3² M1 A1
Answer: 2 × 2 × 3 × 3 (or 22 × 32)
Express 56 as the product of its prime factors.
Worked Solution & Answer
56 = 8 × 7 = 2 × 2 × 2 × 7 (or 2³ × 7) M1 A1
Answer: 2 × 2 × 2 × 7
Write 60 as a product of its prime factors in index form.
Worked Solution & Answer
60 = 2 * 30 = 2 * 2 * 15 = 2² * 3 * 5. M2 A1
Answer: 2² × 3 × 5
Write 36 as a product of prime factors.
Give your answer in index form.
Give your answer in index form.
Answer
Worked Solution & Answer
36 = 2 × 18 = 2 × 2 × 9 = 22 × 32 M2 A1
Answer: Answer 22 × 32
Write 200 as a product of prime factors.
Give your answer in index form.
Give your answer in index form.
Answer
Worked Solution & Answer
200 = 2 × 100 = 2 × 2 × 50 = 2 × 2 × 2 × 25 = 23 × 52 M2 A1
Answer: Answer 23 × 52
Work out
(a) Express 250 as a product of its prime factors. (2)
(b) Find the lowest common multiple (LCM) of 30 and 25 (2)
(a) Express 250 as a product of its prime factors. (2)
(b) Find the lowest common multiple (LCM) of 30 and 25 (2)
(a) Express 250 as a product of its prime factors. [2 marks]
(b) Find the lowest common multiple (LCM) of 30 and 25 [2 marks]
Worked Solution & Answer
(a)
250 = 2 × 125 = 2 × 5³ M1 A1
Answer (a): 2 × 53
(b)
30 = 2 × 3 × 5 M1
25 = 5²
LCM = 2 × 3 × 5² = 150 A1
25 = 5²
LCM = 2 × 3 × 5² = 150 A1
Answer (b): 150
Work out
(a) Write 156 as a product of its prime factors. (2) (b) Find the highest common factor (HCF) of 156 and 130 (2)
(a) Write 156 as a product of its prime factors. (2) (b) Find the highest common factor (HCF) of 156 and 130 (2)
(a) Write 156 as a product of its prime factors. [2 marks]
(b) Find the highest common factor (HCF) of 156 and 130 [2 marks]
Worked Solution & Answer
(a)
156 = 2 × 78 = 2 × 2 × 39 = 2² × 3 × 13 M1 A1
Answer (a): 22 × 3 × 13
(b)
130 = 2 × 5 × 13 M1
HCF = 2 × 13 = 26 A1
HCF = 2 × 13 = 26 A1
Answer (b): 26
Gregor owns a restaurant.
(a) The diagram shows a circular place mat. [2 marks]
The radius of the circular place mat is 14 cm.
Calculate the circumference of the circular place mat.
Calculate the circumference of the circular place mat.
(b) Gregor plans to buy some spoons and forks.
A bag of 24 spoons costs £19.95.
A box of 18 forks costs £15.55.
Bags and boxes cannot be split.
Gregor decides to buy the same number of spoons as forks.
He places an order to buy the smallest number of each that he can.
Complete the details on the order form shown below.
You must show all your working.
@@TABULAR0@@ [5 marks]
A bag of 24 spoons costs £19.95.
A box of 18 forks costs £15.55.
Bags and boxes cannot be split.
Gregor decides to buy the same number of spoons as forks.
He places an order to buy the smallest number of each that he can.
Complete the details on the order form shown below.
You must show all your working.
@@TABULAR0@@ [5 marks]
Worked Solution & Answer
(a)
C = 2 * π * r = 2 * π * 14 = 28π ≈ 87.96 cm (or 2 * 3.14 * 14 = 87.92 cm). M1 A1
Answer (a): 87.96 cm (or 88 cm, 87.92 cm)
(b)
Smallest common multiple of 24 and 18 is 72 M1
Bags of spoons = 72 / 24 = 3 bags M1
Cost of spoons = 3 * 19.95 = £59.85 M1
Boxes of forks = 72 / 18 = 4 boxes M1
Cost of forks = 4 * 15.55 = £62.20
Total cost = 59.85 + 62.20 = £122.05. A1
Bags of spoons = 72 / 24 = 3 bags M1
Cost of spoons = 3 * 19.95 = £59.85 M1
Boxes of forks = 72 / 18 = 4 boxes M1
Cost of forks = 4 * 15.55 = £62.20
Total cost = 59.85 + 62.20 = £122.05. A1
Answer (b): 3 bags of spoons (£59.85), 4 boxes of forks (£62.20), Total = £122.05