AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 1
3 marks · Easy difficulty · Short Answer
Find the highest common factor of 16 and 24, the lowest common multiple of 10 and 15, and express 42 as a product of prime factors.
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Mark scheme
Show the mark scheme
How to answer it
HCF, LCM and Prime Factor Decomposition
📋 What this question tests
This question assesses fundamental number skills essential across both Foundation and Higher tiers:
- Finding the Highest Common Factor (HCF) of two positive integers.
- Finding the Lowest Common Multiple (LCM) of two positive integers.
- Expressing an integer as a product of prime factors using prime factor trees or successive division.
Question 1 (a)
Work out the highest common factor (HCF) of 16 and 24 [1 mark]
✅ Correct Answer
8
Mark scheme: [1 mark] B1 for 8
📐 Step-by-Step Method
Step 1: List all factors of 16:
1, 2, 4, 8, 16
1, 2, 4, 8, 16
Step 2: List all factors of 24:
1, 2, 3, 4, 6, 8, 12, 24
1, 2, 3, 4, 6, 8, 12, 24
Step 3: Identify common factors:
1, 2, 4, 8
1, 2, 4, 8
Step 4: Select the highest: 8
💡 Key Knowledge
- A factor is an integer that divides exactly into another number with no remainder.
- HCF is the largest factor shared by both numbers.
- For small numbers, listing factor pairs is often quicker than a Venn diagram.
❌ Common Errors
- Confusing HCF with LCM: Writing a multiple (e.g. 48) instead of a factor.
- Stopping too early: Writing a common factor that is not the highest, such as 2 or 4.
Question 1 (b)
Work out the lowest common multiple (LCM) of 10 and 15 [1 mark]
✅ Correct Answer
30
Mark scheme: [1 mark] B1 for 30
📐 Step-by-Step Method
Step 1: List the multiples of 10 (times table):
10, 20, 30, 40, 50, ...
10, 20, 30, 40, 50, ...
Step 2: List the multiples of 15:
15, 30, 45, 60, ...
15, 30, 45, 60, ...
Step 3: Identify the smallest number appearing in both lists: 30
💡 Key Knowledge
- A multiple is the result of multiplying a number by an integer (think times tables).
- LCM is the smallest positive integer divisible by both given numbers.
- Multiples are always equal to or larger than the original numbers (unless 0).
❌ Common Errors
- Giving a factor instead: Answering 5 (which is the HCF of 10 and 15, not the LCM).
- Multiplying without checking for smaller multiples: Giving 150 (10 × 15). 150 is a common multiple, but not the lowest.
Question 1 (c)
Write 42 as a product of its prime factors. [1 mark]
✅ Correct Answer
2 × 3 × 7
(Accepted in any order: e.g. 7 × 2 × 3)
Mark scheme: [1 mark] B1 for 2 × 3 × 7
📐 Step-by-Step Method
Step 1: Start by dividing 42 by the smallest prime number (2):
42 ÷ 2 = 21 (circle 2)
42 ÷ 2 = 21 (circle 2)
Step 2: 21 is not divisible by 2, try 3:
21 ÷ 3 = 7 (circle 3)
21 ÷ 3 = 7 (circle 3)
Step 3: 7 is already a prime number (circle 7).
Step 4: Write the circled primes multiplied together:
2 × 3 × 7
2 × 3 × 7
🧠 Exam Technique & Examiner Notes
- "Product" means multiply: You must write multiplication signs ( × ). Dots ( 2 · 3 · 7 ) are condoned.
- Writing a list loses the mark: Simply writing 2, 3, 7 or leaving prime factors circled on a tree diagram without writing a multiplication string scores 0 marks.
- Never include 1: 1 is not a prime number. Including × 1 gets 0 marks.
- Quick check: Multiply your factors back together ( 2 × 3 = 6 , 6 × 7 = 42 ) to make sure you get the starting number!
❌ Common Errors
- Writing a sum instead of a product: 2 + 3 + 7 (scores 0).
- Including non-prime numbers: e.g. 6 × 7 (6 is composite, not prime).
- Including the number 1: 1 × 2 × 3 × 7 (scores 0).
Topics
Number · 3.1.1 Structure and calculation
Question and mark scheme from the AQA GCSE Mathematics examination, Paper 2 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.