Step by step
The question
Question: find the least common multiple of 8, 12 and 15.
The multiples of 8 are 8 times 1, 2, 3 and so on.
The least common multiple, the LCM, is the smallest number in all the lists.
1 Find the prime factors
8 is even, so 2 goes in.
8 is 2 times 4.
4 is even, so 2 goes in.
4 is 2 times 2.
2 is prime, so we stop.
12 is even, so 2 goes in.
12 is 2 times 6.
6 is even, so 2 goes in.
6 is 2 times 3.
3 is prime, so we stop.
3 goes into 15.
15 is 3 times 5.
5 is prime, so we stop.
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.
8 has the most 2s: three.
We take all three.
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.
2 times 2 is 4.
4 times 2 is 8.
8 times 3 is 24.
24 times 5 is 120.
So the LCM is 120.
Careful: every number goes into the LCM.
So it is never smaller than the biggest one.
3 Check
Let's check: each number goes into 120.
No numbers are left over.
So 120 is right!
Let's look back.
First: find the prime factors.
Then: multiply each prime the most times.
Last: check.