Year 7 Maths • AC9M7N02
Prime Factorisation, Prime Powers, HCF & LCM
Show prime-factor working clearly. Use exponent notation where appropriate.
Part A
Five short multistep problems
Let N = 2³ × 3². (a) Write N in standard form. (b) List all factors of N that are multiples of 12.
48 = 2⁴ × 3 and 180 = 2² × 3² × 5. (a) Find the HCF. (b) Use it to simplify 48:180.
A number is divisible by both 18 and 30. (a) Use prime factors to find the LCM. (b) Write the first two positive multiples of that LCM.
Write 360 as a product of powers of primes. Then use the exponents to determine the number of positive factors of 360.
N = 2⁴ × 5². (a) Write N in standard form. (b) Decide whether N is divisible by 40 and justify using prime exponents.
Part B
Five deep multistep problems
A = 2⁵ × 3² × 7 and B = 2³ × 3³ × 5. Find HCF(A,B) and LCM(A,B). Then explain why HCF × LCM = A × B for these two numbers.
Find the smallest positive integer divisible by 84, 90 and 126. Prime-factorise each number first, then justify why the LCM uses the highest power of each prime.
A number N is divisible by 72, has exactly three distinct prime factors and is less than 1000. Prime-factorise 72, identify possible third primes, and give at least four possible values of N.
X = 2ᵃ × 3ᵇ × 5ᶜ has exactly 36 positive factors, where a,b,c are positive integers. Use (a+1)(b+1)(c+1)=36 to find all exponent triples up to order, then give one possible value of X.
A = 2⁴ × 3 × 7. Construct three different values of B such that HCF(A,B)=84. For each B, show its prime factorisation and verify the HCF.
Part C • Gifted investigation
How does prime structure classify numbers?
Investigate how prime factorisation describes the structure of a number rather than just its size. Compare families such as 2ᵃ3ᵇ; analyse how HCF and LCM change when exponents change; and research, at an introductory level, why large-prime factorisation matters in RSA encryption. Present a one-page report with at least three worked numerical examples and a clear distinction between what Year 7 prime factorisation can explain and what advanced cryptography adds.