1. Which number is prime?
2. Which number is divisible by 6?
3. What is the prime factorisation of 360?
4. What is HCF(18, 24)?
5. Two correct factor trees start differently. What must be true?
6. Use divisibility tests to decide whether 7,452 is divisible by 2, 3, 4, 5, 6 and 10.
7. Write 756 as a product of prime powers.
8. Find the HCF and LCM of 48 and 180 using prime factors.
9. Explain why 1 is not prime.
Answers and reasoning
1. Answer
29. 29 has no positive factors other than 1 and 29.
2. Answer
234. 234 is even and its digit sum is 9, so it is divisible by both 2 and 3.
3. Answer
2³ × 3² × 5. 360 = 8 × 9 × 5 = 2³ × 3² × 5.
4. Answer
6. 18 = 2 × 3² and 24 = 2³ × 3, so the common minimum powers give 2 × 3 = 6.
5. Answer
Their prime factors match after reordering. The fundamental theorem of arithmetic gives a unique prime factorisation, apart from order.
6. Model response
It is divisible by 2 (even), 3 (digit sum 18), 4 (52 divisible by 4) and 6 (both 2 and 3). It is not divisible by 5 or 10.
7. Model response
756 = 2² × 3³ × 7. Check: 4 × 27 × 7 = 756.
8. Model response
48 = 2⁴ × 3; 180 = 2² × 3² × 5. HCF = 2² × 3 = 12. LCM = 2⁴ × 3² × 5 = 720.
9. Model response
A prime must have exactly two positive factors. The number 1 has only one positive factor, so it is neither prime nor composite.