Primes, HCF and LCM

Foundation and Higher

The idea

Every whole number above $1$ is a product of primes in exactly one way: $60 = 2^2 \times 3 \times 5$ . The HCF is the biggest number that divides both; the LCM is the smallest number both divide into.

Write each number as a product of primes with a factor tree.
HCF: multiply the primes the numbers share, each to the lower power.
LCM: multiply every prime that appears, each to the higher power.

Watch out Things that happen together again (buses, lights, packs) need the LCM; splitting into equal groups needs the HCF.

See it worked

Worked exampleNon-calculator4 marks

Answer each part without a calculator.

(a) Write down all the factors of $18$ .

(b) Write down the prime numbers in this list: $21$ , $23$ , $25$ , $27$ , $29$ .

(c) Write $60$ as a product of its prime factors.

(a) Find the factor pairs: $1 \times 18$ , $2 \times 9$ , $3 \times 6$ .
(b) $21 = 3 \times 7$ , $25 = 5 \times 5$ and $27 = 3 \times 9$ have other factors. $23$ and $29$ have exactly two factors, 1 and themselves.
(c) Split into factors until every branch ends in a prime. $60 = 2 \times 30 = 2 \times 2 \times 15 = 2 \times 2 \times 3 \times 5$
Written with a power: $60 = 2^2 \times 3 \times 5$ .

Answer (a) $1, 2, 3, 6, 9, 18$ ; (b) $23$ and $29$ ; (c) $2^2 \times 3 \times 5$

Your turn

Question 1Non-calculator3 marks

Find the highest common factor of 42 and 70.

Question 2Non-calculator3 marks

Two lights flash together. One then flashes every 18 seconds and the other every 24 seconds. How many seconds pass before they next flash together?

Question 3Non-calculator6 marks

$A = 2^3 \times 3^2 \times 5$ and $B = 2^2 \times 3^4 \times 7$

(a) Find the highest common factor of $A$ and $B$ .

(b) Find the lowest common multiple of $A$ and $B$ . You may leave your answer as a product of powers of prime factors.

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