The greatest common divisor (GCD) is the largest number that divides both numbers exactly, and the least common multiple (LCM) is the smallest number that is a multiple of both.
Lay out the multiples of 12 and of 18 on number lines and they meet at 36 and 72; the first meeting point, 36, is the LCM.
Split both numbers into prime factors and sort them into a Venn diagram: the product of the overlap is the GCD, and the product of everything is the LCM.
On this page you change the two numbers to see where their multiples meet and which prime factors they share, and why the product of the two numbers equals GCD × LCM.
Where the multiples meet
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Number lines of the multiples of two numbers: gold dots are common multiples, and the first match is the LCM
12
18
👀 Where the multiples meet
①List the multiples of each number and some numbers appear in both lists. These are common multiples
②The smallest common multiple is the LCM. In the picture it is the first place where a dashed line joins the two rows
③Every common multiple is a multiple of the LCM. For 12 and 18 they go 36, 72, 108, going up by 36 each time
④If one number is a multiple of the other, the larger number is itself the LCM
Prime factors in a Venn diagram
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Venn diagram of prime factors: the middle holds the prime factors both numbers share
🔍 Sorting into a Venn diagram
①12 = 2² × 3 and 18 = 2 × 3². Put the 2 and 3 that both share in the middle, the leftover 2 on the 12 side and the leftover 3 on the 18 side
②Multiply what is in the middle to get the GCD: 2 × 3 = 6
③Multiply everything in the left, middle and right to get the LCM: 2 × 2 × 3 × 3 = 36
④If no prime factor is shared, the GCD is 1, and the two numbers are called coprime
GCD × LCM = the product of the two numbers
Link with the product
A × B = GCD × LCM
12 × 18 = 216 = 6 × 36
Finding the LCM
LCM = A × B ÷ GCD
Once you have the GCD, the LCM follows straight away: 216 ÷ 6 = 36
💡 Why the products match
①Written as prime factors, A × B multiplies the middle of the Venn diagram twice
②The LCM multiplies each of the three parts once, and the GCD is the middle part
③So GCD × LCM also multiplies the middle twice and the sides once each, which is exactly A × B
④Use this to check your answers once you have worked them out
Check with examples
Example 1
Find the GCD of 12 and 18.
1
Write each number as a product of prime factors.
12 = 2² × 3, 18 = 2 × 3²
2
Take the shared prime factors 2 and 3, each with the smaller power, and multiply.
GCD = 2 × 3 = 6
▸ 6
It is the product of the 2 and 3 in the middle of the Venn diagram.
Example 2
Find the LCM of 12 and 18.
1
Write each number as a product of prime factors.
12 = 2² × 3, 18 = 2 × 3²
2
Take every prime factor, each with the larger power, and multiply.
LCM = 2² × 3² = 36
▸ 36
Check: 12 × 18 = 216 and 6 × 36 = 216. On the number lines, too, the first meeting point is 36.
Exam checklist
GCD
Multiply the shared prime factors, each with the smaller power
The largest number that divides both numbers exactly
LCM
Multiply every prime factor, each with the larger power
The smallest number that is a multiple of both
Grade 7 school test style
A 12 cm by 18 cm rectangle is to be covered exactly with equal square tiles, as large as possible. How long is a side of each tile?
①2
②3
③6
④9
⑤12
▸ ③ 6
1
The side of a tile must divide both 12 and 18 exactly, and be the largest such number.
side = GCD of 12 and 18
2
Find the GCD.
= 2 × 3 = 6 (cm)
🎯 Exam checklist
①Problems about splitting things as largely as possible, like the biggest square tile, use the GCD
②Problems about when things first line up again, like buses leaving together, use the LCM
③The GCD and LCM of three numbers are found the same way
④Two numbers whose GCD is 1 are coprime, and then their LCM is their product