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Year 7AC9M7N02

Prime Factorisation, HCF and LCM

Represent natural numbers as products of powers of prime numbers using exponent notation.

Every number breaks down into primes in exactly one way. Once you have that breakdown, highest common factor and lowest common multiple fall straight out of it.

Method

A prime has exactly two factors
2, 3, 5, 7, 11, 13…2,\ 3,\ 5,\ 7,\ 11,\ 13\ldots
1 is not prime — it has only one factor.
HCF: take the shared primes
Multiply the primes both numbers have, using the lower power of each.
LCM: take every prime
Use the higher power of each prime that appears in either number.

Worked example

e.g. Find the HCF and LCM of 36 and 48.\text{Find the HCF and LCM of } 36 \text{ and } 48.
  1. Break the first number into primes using a factor tree.36=22×3236 = 2^2 \times 3^2
  2. Do the same for the second number.48=24×348 = 2^4 \times 3
  3. For the HCF, take each shared prime to its lower power.22×3=122^2 \times 3 = 12
  4. For the LCM, take each prime to its higher power.24×32=1442^4 \times 3^2 = 144
  5. Check: the HCF divides both, and both divide the LCM.36÷12=3144÷48=3✓36 \div 12 = 3 \quad 144 \div 48 = 3 \quad\checkmark

Practice

Write each as a product of primes first.

1
Write 60 as a product of primes.\text{Write } 60 \text{ as a product of primes.}
Answer22×3×52^2 \times 3 \times 5
2
HCF of 24 and 36\text{HCF of } 24 \text{ and } 36
Answer1212
3
LCM of 6 and 8\text{LCM of } 6 \text{ and } 8
Answer2424
4
HCF of 15 and 28\text{HCF of } 15 \text{ and } 28
Answer1 — they share no prime1 \text{ — they share no prime}
5
Is 51 prime?\text{Is } 51 \text{ prime?}
AnswerNo: 51=3×17\text{No: } 51 = 3 \times 17
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