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 Math Help > Procedures > LCM and GCD

# Least Common Multiple (LCM) and Greatest Common Divisor (GCD)

 Please note: Sometimes LCM is called LCD, for Lowest Common Denominator.  This is because a common denominator is a common multiple of the original denominators.  But this use of "D" confuses people who might think it stands for "Divisor". Sometimes GCD is called GCF, for Greatest Common Factor.  A Factor is the same thing as a Divisor. Just understand that these concepts go by various names, so don't be surprised if you see one of their aliases.

On 10/17/01 11:37:46 PM, Ashley Martin wrote:

"greatest common multiples and least common ' '"

>i just dont know how to do them i forgot!!
>(least and greatest) please help
>please help!!!
>somebody reply

There is no greatest multiple, common or otherwise. You probably mean the Least Common Multiple. And you mean Greatest Common Divisor (or Factor).

Let's take them one at a time.

To find the LCM of a number, first find the prime factorization of the number. Write it as the product of powers of prime numbers. Then, for each prime number present in any of the prime factorizations, write it with the highest factor that appears in any one of the prime factorizations. This is the LCM.

Let's do an example. Find the LCM of 40, 44, and 45.

40 = 23 30 51 110
44 = 22 30 50 111
45 = 20 32 51 110

Now take the maximum of each exponent:

LCM = 23 32 51 111

Multiply it out, and you'll see

LCM = 8*9*5*11 = 3960

Now, let's look at the GCD. It's just like the LCM, except instead of taking the maximum of each exponent, you take the minimum.

We'll find the GCD of 360, 300, and 375.

360 = 23 32 51
300 = 22 31 52
375 = 20 31 53
---------------------
GCD = 20 31 51

GCD = 1*3*5 = 15

Do you get it?

### Related pages in this website

Common Denominator answers the question, "Why do you need a common denominator when adding fraction but not when multiplying?"

The webmaster and author of this Math Help site is Graeme McRae.