Step by step: way 1, lists
The question
Question: find the least common multiple of 6 and 10.
LCM(6, 10) = ?
The multiples of 6 are 6 times 1, 2, 3 and so on.
The least common multiple, the LCM, is the smallest number in both lists.
multiples of 6: 6, 12, 18, ...
LCM: the smallest common one
1 Count up
We count up in 6s: 6, 12, 18, 24 and 30.
In 6s:6, 12, 18, 24, 30
Now in 10s: 10, 20 and 30.
In 10s:10, 20, 30
30 is the first number in both lists.
So the LCM is 30.
LCM(6, 10) = 30
Careful: every number goes into the LCM.
So it is never smaller than the biggest one.
Careful: LCM ≥ the biggest number
2 Check
Let's check: each number goes into 30.
30 ÷ 6 = 5
30 ÷ 10 = 3
Nothing is left over.
So 30 is right!
Let's look back.
First: count up.
Last: check.
Step by step: way 2, prime factors
The question
Question: find the least common multiple of 6 and 10.
LCM(6, 10) = ?
The multiples of 6 are 6 times 1, 2, 3 and so on.
The least common multiple, the LCM, is the smallest number in both lists.
multiples of 6: 6, 12, 18, ...
LCM: the smallest common one
1 Find the prime factors
6 is even, so 2 goes in.
6 is 2 times 3.
3 is prime, so we stop.
6 = 2 × 3
10 is even, so 2 goes in.
10 is 2 times 5.
5 is prime, so we stop.
10 = 2 × 5
2 Multiply each prime the most times
The LCM must be a multiple of every number.
So it needs each prime as many times as it shows up the most.
2 shows up once at most.
We take one 2.
3 shows up once at most.
We take one 3.
5 shows up once at most.
We take one 5.
We multiply the primes we took.
LCM(6, 10) = 2 × 3 × 5
2 times 3 is 6.
6 times 5 is 30.
2 × 3 = 6
6 × 5 = 30
So the LCM is 30.
= 30
Careful: every number goes into the LCM.
So it is never smaller than the biggest one.
Careful: LCM ≥ the biggest number
3 Check
Let's check: each number goes into 30.
30 ÷ 6 = 5
30 ÷ 10 = 3
Nothing is left over.
So 30 is right!
Let's look back.
First: find the prime factors.
Then: multiply each prime the most times.
Last: check.
