Prime factorisation, HCF and LCM

Keep breaking composite factors

72 = 8 × 9 = 2³ × 3². A prime factorisation ends in primes. An alternative factor tree gives the same prime factors in a different order. Check by recombining: 8 × 9 = 72.

For 72 and 120, write 72 = 2³ × 3² and 120 = 2³ × 3 × 5. Their HCF uses only common primes with the smaller exponents: 2³ × 3 = 24. Their LCM uses every prime with the larger exponent: 2³ × 3² × 5 = 360. Check that both numbers divide 360 exactly.

A perfect square has even exponents in its prime factorisation; a perfect cube has exponents divisible by 3. Thus square root of 144 = square root of (2⁴ × 3²) = 2² × 3 = 12 and cube root of 216 = cube root of (2³ × 3³) = 6.

Choose the relevant common quantity

Two lights flash every 18 s and 24 s and flash together now. When do they next flash together?

Show answer

18 = 2 × 3² and 24 = 2³ × 3, so LCM is 2³ × 3² = 72 s. A common factor would answer a different grouping question.

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