A divisor of a number divides it with nothing left over, and a multiple of a number is that number taken 1, 2, 3 or more times.
The divisors of 12 stop at six numbers, 1, 2, 3, 4, 6 and 12, but the multiples of 12 go on forever: 12, 24, 36 and so on.
The largest divisor two numbers share is their greatest common divisor, and the smallest multiple they share is their least common multiple.
Choose two numbers below and the two pictures show the divisors they share and the first multiple where they meet.
Divisors and multiples
Divisors and multiples answer two everyday kinds of question: sharing out and lining up. To share 12 sweets and 18 chocolates equally with nothing left over, the number of children has to divide both amounts, so it must be a common divisor, and the most children you can share with is the greatest one, 6. If one bus leaves every 12 minutes and another every 18 minutes and they have just left together, they leave together again at every common multiple, and the first time is after 36 minutes, the least common multiple. Deciding first whether a problem is about sharing out or about meeting again tells you whether you are looking for divisors or multiples.
🍰 If it shares out equally, it is a divisor
①Share 12 pieces of cake equally among 3 people and each gets 4. Because 12 ÷ 3 = 4 with nothing left over, 3 is a divisor of 12 and 12 is a multiple of 3
②The only numbers that divide 12 exactly are 1, 2, 3, 4, 6 and 12, so 12 has six divisors
③1 and the number itself are always divisors. Finding divisors in pairs that multiply to the number, like 1 × 12, 2 × 6 and 3 × 4, makes sure you miss none
Common divisors and the GCD
12
18
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Divisors of 12 and of 18, overlapped in a Venn diagram
The blue circle holds the divisors of the first number and the orange circle holds the divisors of the second. The green numbers in the overlap are the common divisors, the ones both numbers share. With the starting values 12 and 18, only 4 and 12 stay on the left, only 9 and 18 stay on the right, and 1, 2, 3 and 6 sit in the middle. The largest of these, 6, is written at the bottom as the greatest common divisor: it is the biggest number that divides both 12 and 18 exactly. The number 1 divides everything, so it always lands in the middle, and when one number divides the other, every divisor of the smaller one moves into the overlap.
🔍 Common divisors and the GCD
①Divisors that two numbers share are called common divisors. In the Venn diagram they are the numbers in the overlap
②The largest common divisor is the greatest common divisor (GCD)
③Every common divisor is a divisor of the GCD: the common divisors of 12 and 18, namely 1, 2, 3 and 6, are exactly the divisors of 6
Common multiples and the LCM
The most common slip is swapping the greatest common divisor and the least common multiple. A divisor can never be bigger than its number, so the GCD can never be bigger than the smaller of the two numbers; a multiple can never be smaller than its number, so the LCM can never be smaller than the larger one. If you work out 12 and 18 and get a GCD of 36, bigger than 18, or an LCM of 6, smaller than 12, you have swapped them. After answering, check that 12 × 18 = 216 equals GCD × LCM = 6 × 36 = 216, which also catches arithmetic mistakes.
↔️ The same two numbers, now with multiples
The two numbers you chose above also drive the number lines below. The top row shows multiples of the first number and the bottom row multiples of the second, on the same scale, and a green dashed line joins the first value the two rows share.
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Multiples of 12 and of 18 on number lines with the same scale
🎯 Common multiples and the LCM
①Multiples that two numbers share are called common multiples, and the smallest one is the least common multiple (LCM)
②The green dot where the two rows first reach the same value is the LCM. For 12 and 18 it is 36
③Every common multiple is a multiple of the LCM, so they go 36, 72, 108 and so on
④The product of two numbers equals the product of their GCD and LCM: 12 × 18 = 216 = 6 × 36
Summary
Divisors and multiples
If A × n = B, then A is a divisor of B and B is a multiple of A
3 × 4 = 12, so 3 and 4 are divisors of 12, and 12 is a multiple of both 3 and 4
GCD and LCM
A × B = GCD(A, B) × LCM(A, B)
12 × 18 = 216 and 6 × 36 = 216
🎯 At a glance
①Divisor: a number that divides another exactly, and there are only so many
②Multiple: a number taken 1, 2, 3 or more times, and they never end
③GCD: the largest common divisor
④LCM: the smallest common multiple
⑤Product of two numbers = GCD × LCM
Examples and Unit Test
Example 1
Find all the divisors of 12.
1
Pair up numbers that multiply to 12: 1 × 12, 2 × 6 and 3 × 4.
2
List the numbers from these pairs in order: 1, 2, 3, 4, 6, 12.
▸ 1, 2, 3, 4, 6, 12
After 3 × 4 the next pair would start with 4, which you already have, so you can stop there.
Example 2
Find the GCD of 12 and 18.
1
The divisors of 12 are 1, 2, 3, 4, 6 and 12; the divisors of 18 are 1, 2, 3, 6, 9 and 18.
2
The numbers on both lists are 1, 2, 3 and 6, and the largest is 6.
▸ 6
This is the starting setting of the first picture: 1, 2, 3 and 6 sit in the overlap and the GCD, 6, is written at the bottom.
Unit Test
What is the LCM of 4 and 6?
①2
②12
③24
④10
▸ ② 12
1
Multiples of 4: 4, 8, 12, 16, … Multiples of 6: 6, 12, 18, …
2
The first number on both lists is 12, so the LCM is 12.