From the prime factorisations of two numbers, the HCF is the product of the shared primes at their smallest powers, and the LCM is the product of all the primes at their largest powers.
The highest common factor (HCF) of two numbers is the largest whole number that divides both of them exactly. The lowest common multiple (LCM) is the smallest positive whole number that both of them divide exactly.
Listing all the factors or multiples works for small numbers but takes many steps for large ones. Prime factorisation gives a method whose length depends only on how many primes are involved.
Take 6 and 20. Their prime factorisations are 6 = 2 × 3 and 20 = 2² × 5.
The HCF is the product of the primes that appear in both factorisations, each raised to the smaller of its two powers. The prime 2 appears in both: once in 6 and twice in 20. The smaller power is 2¹. The prime 3 appears only in 6, and 5 only in 20, so neither can be part of a number that divides both.
HCF(6, 20) = 2¹ = 2
Why is the smaller power of each shared prime used for the HCF?
The HCF must divide both numbers. 6 contains the prime 2 only once. So any number that divides 6 can contain 2 at most once. That 20 contains 2 twice does not help, because the factor must also divide 6.
With 2² = 4, the result would no longer divide 6. The number with fewer copies of a prime sets the limit.
The LCM is the product of every prime that appears in either factorisation, each raised to the larger of its powers.
For 6 and 20, the primes are 2, 3 and 5. The largest powers are 2² (from 20), 3¹ (from 6) and 5¹ (from 20).
LCM(6, 20) = 2² × 3 × 5 = 60
The LCM must be a multiple of both numbers. So it must contain each prime at least as many times as either number contains it. That is why the larger power is used.
One set of prime factors, read two ways. The HCF uses only the primes both numbers share, at the smaller power. The LCM uses every prime, at the larger power.
Take 96 and 404. 96 = 2⁵ × 3 and 404 = 2² × 101.
The only shared prime is 2, and the smaller power is 2². So HCF = 4. For the LCM, take 2⁵, 3 and 101: LCM = 2⁵ × 3 × 101 = 32 × 303 = 9696.
Factorising each number once gave both answers. Listing the multiples of 404 to find the LCM would take 24 steps.
Find the HCF and LCM of 12 and 18 from their prime factorisations.
12 = 2² × 3 and 18 = 2 × 3².
HCF: both contain 2 and 3. Smaller powers: 2¹ and 3¹. HCF = 2 × 3 = 6.
LCM: larger powers: 2² (from 12) and 3² (from 18). LCM = 4 × 9 = 36.
The method extends directly. For the HCF, use only the primes that appear in every number, each at its smallest power. For the LCM, use every prime that appears in any of the numbers, each at its largest power.
For the HCF, a prime shared by only two of three numbers does not count.
1To find the HCF of two numbers from their prime factorisations, you take…
2To find the LCM, you take…
3With 12 = 2² × 3 and 18 = 2 × 3², the HCF of 12 and 18 is ____.
4With 12 = 2² × 3 and 18 = 2 × 3², the LCM is…
5Why does the HCF use the smaller power of each shared prime?
6For 96 = 2⁵ × 3 and 404 = 2² × 101, the HCF is…
Next: the relation HCF × LCM = product of the two numbers.
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