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LCM (Least Common Multiple) and
GCF/GCD (Greatest Common Factor/Divisor)
To find either the Least Common Multiple (LCM) or Greatest Common Factor (GCF/GCD) of two
numbers, you always start out the same way: you find the prime fctorizations of the two numbers.
Then you put the factors into a nice neat grid of rows and columns, and then compare and
contrast and take what you need. Here's how it works:

Find the GCF/GCD and LCM of
2940 and 3150.
First, you factor:
Then the factorizations are: Copyright © Elizabeth Stapel 2000-2006 All Rights Reserved
2940 = 2 × 2 × 3 × 5 × 7 × 7
3150 = 2 × 3 × 3 × 5 × 5 × 7
Now write these factors out all nice and neat, with the factors lined up according to
occurrance:
Note how the factors are listed very orderly. This orderly listing will do most of the work
for you!
The GCF/GCD is the biggest number that will divide into both 2940 and 3150. In other
words, it's the number that contains all the common factors. So the GCF/GCD is the
product of any and all factors that 2940 and 3150 share. Looking at the nice neat listing,
you can see that the numbers both have a factor of 2; 2940 has a second copy, but
3150 does not, so you can only count the one copy toward your GCF/GCD. The numbers
also share one copy of 3, one copy of 5, and one copy of 7.
Then the GCF/GCD is 2
× 3 × 5 × 7 = 210.
On the other hand, the LCM is the smallest number that both 2940 and 3150 will divide
into. That is, it is the smallest number that contains both 2940 and 3150, that both
numbers fit in to. Then it will be the smallest number that contains one of every factor in
these two numbers. Looking back at the listing, you'll see that 3150 has one copy of the
factor of 2; 2940 has two copies. Since the LCM must contain all factors of each number,
the LCM must contain both copies of 2. However, to avoid overduplication, the LCM does
not need three copies, because neither 2940 nor 3150 contains three copies.
This fact often causes confusion, so let's spend a little extra time on this. Consider two
smaller numbers, 4 and 8, and their LCM. The number 4 factors as 2 × 2; 8 factors as 2
× 2 × 2. The LCM needs only have three copies of 2, in order to be divisible by both 4
and 8. That is, the LCM is 8. You do not need to take the three copies of 2 from the 8,
and then throw in two extra copies from the 4. This would give you 32. While 32 is a
common multiple, because 4 and 8 both divide evenly into 32, 32 is not the LEAST
(smallest) common multiple, because you over-duplicated the 2s when you threw in the
extra copies from the 4. Again, let the nice neat listing keep track of things when the
numbers get big.
So, the LCM of 2940 and 3150 must contain both copies of the factor 2. By the same
reasoning, the LCM must contain both copies of 3, both copies of 5, and both copies of
7:
Then the LCM is 2
× 2 × 3 × 3 × 5 × 5 × 7 × 7 = 44,100.
Using this "factor" method of listing the prime factors neatly in a table, you can always easily find
the LCM and GCF/GCD. Completely factor the numbers you are given, list the factors neatly, with
only one factor for each column (you can have a 2s column, a 3s column, etc, but a 3 would
never go in a 2s column), and then carry the needed factors down to the bottom row. For the
GCF/GCD, you carry down only those factors that all the listings share; for the LCM, you carry
down all the factors, regardless of how few numbers contained that factor in their listings.

Find the LCM and GCF/GCD of 27,
90, and 84.
First, find the prime factorizations:
Then list things out neatly:
Then the GCF/GCD and the LCM are given by:
Then the GCF/GCD is 3 and the LCM is 3,780.

Find the GCF/GCD and LCM of 3, 6, and 8.
First factor and list:
Then the GCF/GCD and LCM are given by:
Note that 3, 6, and 8 share no common factors. While 3 and 6 share a factor, and 6 and
8 share a factor, there is no prime factor that all three of them share. Since 1 divides into
everything, then the greatest common factor in this case is just 1. When 1 is the
GCF/GCD, the numbers are said to be "relatively" prime; that is, they are prime, relative
to each other.
Then the GCF/GCD is 1 and the LCM is 2
× 2 × 2 × 3 = 24.