Stage 8 - Q1 - 2 marks
Write as a product of prime factors.
Hint
Keep dividing composite factors until every factor is prime.
Stage 8 - Q2 - 2 marks
Write as a product of prime factors.
Hint
Keep dividing composite factors until every factor is prime.
Stage 8 - Q3 - 3 marks
Complete the prime factorisation.
Hint
Keep dividing composite factors until every factor is prime.
Stage 8 - Q4 - 3 marks
Complete the prime factorisation.
Hint
Keep dividing composite factors until every factor is prime.
Stage 8 - Q5 - 4 marks
A storage display has 180 counters. Express the total as prime factors, then find the largest power of 2 that divides it.
Hint
Keep dividing composite factors until every factor is prime.
Stage 8 - Q6 - 4 marks
A storage display has 3240 counters. Express the total as prime factors, then find the largest power of 2 that divides it.
Hint
Keep dividing composite factors until every factor is prime.
Stage 8 - Q7 - 5 marks
A number has prime factors 2, 3 and 7, each at least once. It has exactly four prime factors when repeats are counted. Find the smallest possible number.
Hint
Start with one of each required prime. Which extra prime keeps the product smallest?
Stage 8 - Q8 - 5 marks
A pupil says: every multiple of 6 is a multiple of 12. Explain the missing prime-factor condition and give a counterexample.
Hint
Compare the prime powers required by 6 and 12.
Stage 8 - Q9 - 5 marks
Find the smallest positive integer k such that \(600k\) is a cube number.
Hint
A cube groups identical prime factors in threes.
Stage 8 - Q10 - 5 marks
Find the smallest positive integer divisible by 12, 18 and 20 that is also a square number.
Hint
Write each divisor as a product of primes. Which powers must be even?