# Least Common Multiple (or Denominator)

So let’s find the least common multiple (LCM) of 4 and 18. I am now calling the LCD a least common multiple because these numbers are not in a denominator (yet), but it’s the same thing. It’s called a “multiple” because what we are finding is the smallest number that the given numbers both divide into.

So the first step is to factor each of the numbers into its prime factors:

4 = 2 × 2

18 = 2 × 9 = 2 × 3 × 3

The next step to get the LCM (or LCD if the numbers are in a denominator), is to multiply all the prime numbers in the factorisation above, but only the maximum times each appears. So, there are only two factors in the above breakdown, 2 and 3. The “2” appears twice in the “4” but only once in the “18”. The “3” appears twice in the “18. So I will use the “2” only twice and the “3” twice:

LCM = 2 × 2 × 3 × 3 = 4 × 9 = 36

So 36 is the smallest number that 4 and 18 can divide into. Now 4 × 18 = 72 is also a number that they will divide into but 36 is the smallest.

I will use this result in my next post but let’s see some more examples.

What is the LCM of 9 and 24?

9 = 3 × 3

24 = 2 × 2 × 2 × 3

So again only “2” and “3” are present. I will use the “3” only twice in calculating the LCM:

LCM =  2 × 2 × 2 × 3 × 3 = 72

Let’s try finding the LCM of 5 and 15:

5 is already a prime number

15 = 3 × 5

So the LCM = 3 × 5 = 15

Notice that sometimes the LCM is one of the given numbers. Always check to see if one of the numbers divides into the other one. If it does, the bigger number is the LCM.

Let’s try another one. What’s the LCM for 16 and 24:

16 = 2 × 2 × 2 × 2

24 = 2 × 2 × 2 × 3

LCM = 2 × 2 × 2 × 2 × 3 = 48