Before this page: write a number as a product of primes using prime factorisation, and find the highest common factor and the lowest common multiple from index-form factorisations.
Sort two numbers' prime factors into two circles
84 and 90 each get a circle, and each number's prime factors are crossed off its list one at a time. The overlap lies inside both circles, so a prime both numbers contain is written there once and counts for each.
Shared primes are placed first, a pair at a time, until one list has none of that prime left:
Sorting 84 and 90 into a Venn diagram
84 and 90 as circles
84 = 2 × 2 × 3 × 7 and 90 = 2 × 3 × 3 × 5, each listed beside a circle of its own.
Any one of HCF, LCM and the two numbers follows from the other three
84 × 90 = 7,560 counts every region once and the overlap twice: once inside 84 and once inside 90. HCF × LCM = 6 × 1,260 = 7,560 counts the same regions the same way — the overlap once as the HCF and again inside the LCM, and each number's own primes once.
18 = 2 × 3 × 3 and 24 = 2 × 2 × 2 × 3: HCF = 2 × 3 = 6 and LCM = 2 × 2 × 2 × 3 × 3 = 72.
Given 18, the HCF and the LCM, the other number is 6 × 72 ÷ 18 = 24; given both numbers and the HCF, LCM = 18 × 24 ÷ 6 = 72.
HCF × LCM = the product of the two numbers
A third number adds a third circle
For 24, 60 and 90 the centre lies inside all three circles, so a prime all three numbers contain is written there once and counts for each.
Sorting 24, 60 and 90 into a Venn diagram
24, 60 and 90 as circles
24 = 2 × 2 × 2 × 3, 60 = 2 × 2 × 3 × 5 and 90 = 2 × 3 × 3 × 5, each listed beside a circle of its own.
Common mistakes
- Keeping a shared prime in its own region as well as the overlap For 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3, 12's own region keeps one 2 once the other has gone to the overlap. Leaving both 2s in 12's own region as well gives 2 × 3 × 2 × 2 × 3 = 72 for the LCM, twice the correct 36: a prime crossed off both lists is written once.
- Reading the overlap as the LCM 6 is smaller than both 12 and 18, and a multiple of neither. The overlap is the HCF; the LCM, 36, needs every region.
Summary
- One circle per number: a prime on both lists is crossed off both and written once, in the overlap; a prime on one list stays in that number's own region.
- The overlap multiplies to the HCF; every region together multiplies to the LCM.
- For two numbers, HCF × LCM equals their product. Any one of the four values follows from the other three.
- A third number adds a third circle: the centre gives the HCF, and every region together still gives the LCM.
- For three numbers there is no corresponding product identity: factors can be counted a different number of times in the three numbers' product and in HCF × LCM.