Summary:

- Common Multiples must contain all the prime factors of any of the original numbers

- The Least Common Multiple has no spare factors - if you remove one, then it won't be a Common Multiple anymore

- The Least Common Multiple contains only enough factors to match any one original number - but no more.

- The Least Common Multiple is useful as a common denominator when you're adding fractions!

- Common Factors must contain only prime factors that are common to all of the original numbers

- The Greatest Common Factor has no room for more factors - if you add one, then it won't be a Common Factor anymore

- The Greatest Common Factor contains as many factors as it can without making the other numbers jealous

- Common Multiples must contain all the prime factors of any of the original numbers

- The Least Common Multiple has no spare factors - if you remove one, then it won't be a Common Multiple anymore

- The Least Common Multiple contains only enough factors to match any one original number - but no more.

- The Least Common Multiple is useful as a common denominator when you're adding fractions!

- Common Factors must contain only prime factors that are common to all of the original numbers

- The Greatest Common Factor has no room for more factors - if you add one, then it won't be a Common Factor anymore

- The Greatest Common Factor contains as many factors as it can without making the other numbers jealous

## The Math Tutor - Austin

Finding Common Multiples and Factors

Common Multiples:

A "Common Multiple" of a set of numbers is a multiple of each original number - or in other words, it contains each of the original numbers as factors.

It's easy to see if something is a common multiple as long as you split all of the numbers into their prime factors.

Example 1:

140 is a common multiple of 35 and 10, because:

140 has prime factors 2*2*5*7

35 has prime factors 5*7 - and 140 = 2*2*(5*7) includes the factors of 35

10 has prime factors 2*5 - and 140 = 2*(2*5)*7 includes the factors of 10

Example 2:

45 is not a common multiple of 15 and 27, because:

45 has prime factors 3*3*5

27 has prime factors 3*3*3 - and that's one more 3 than 45 has, so 45 does not contain 27 as a factor

Least Common Multiple (LCM):

Common Multiples just have to match all of the prime factors in the original numbers - but the Least Common Multiple (LCM) is the only one to do it with no factors to spare. (That is, if you took any of its factors away, it wouldn't be able to contain the original numbers anymore)

For example - 500, 750, and 50 are all common multiples of 25 and 10. However, of all possible common multiples, 50 is the smallest.

Why so?

Let's look at the prime factors:

25 has factors 5*5

10 has factors 2*5

50 has factors 2*5*5 - it includes the factors of 25 and the factors of 10, so it is a common multiple.

Even more, if we try to take any factors out of 50 to make it smaller, then it will no longer contain either the 25 or the 10:

- if we take a 5 out, then there won't be enough 5's left to match 25

- if we take a 2 out, then we can't match the 10 (which has a 2)

Since 50 has no factors to spare, it is the LCM.

(NOTE that the LCM doesn't have to include the factors of 25 and 10 AT THE SAME TIME)

In fact, it's pretty easy to think of what the LCM has to be if you know the factors of each of the original numbers. You just have to make a number that matches the factors of the original numbers, but has no factors to spare.

Example:

Find the Least Common Multiple (LCM) of 15, 27, and 25, and 45.

15 has prime factors 3*5

27 has prime factors 3*3

25 has prime factors 5*5

45 has prime factors 3*3*5

Our LCM needs:

- two factors of "3" - because some of our numbers have two "3"s, but none of them have more

- two factors of "5" - because one of our numbers has two "5"s, but none of them have more

- no other factors - because none of our numbers have factors other than "3" or "5"

That leaves us with a LCM of 3*3*5*5, or 225.

Common Factors:

A "Common Factor" of a set of numbers is a factor of each of the original numbers - or in other words, it is contained as a factor in each of the original numbers.

Just as with common multiples, it's easy to see if something is a common factor by looking at its prime factors.

Example 1:

18 is a common factor of 90 and 36

18 has prime factors 2*3*3

90 has prime factors (2*3*3)*5, and so it contains (2*3*3), or 18, as a factor

36 has prime factors 2*(2*3*3), so it also contains 18 as a factor.

Example 2:

18 is NOT a common factor of 90 and 135

18 has prime factors 2*3*3

135 has prime factors 3*3*3*5 - since it doesn't have a 2, and 18 does, it cannot contain 18 as a factor!

Greatest Common Factor (GCF):

Common Factors have to fit inside each of the original numbers - but the Greatest Common Factor (GCF) is the absolute biggest one that will fit.

Another way to think of it is this - the GCF is the number with the most factors that won't make any of the original numbers jealous of its factors. (That is - every original number needs to have at least the same factors as the GCF)

It's easy to figure out what the GCF has to be if you know the prime factors of the original numbers.

Example:

Find the Greatest Common Factor (GCF) of 90, 54, and 36.

90 has prime factors 2*3*3*5

54 has prime factors 2*3*3*3

36 has prime factors 2*2*3*3

Our GCF can have:

One "2" - but no more, because otherwise the 90 and the 54 would be jealous (they have only one "2")

Two "3"s - but no more, because otherwise the 90 and the 36 would be jealous

No 5's - because if it did, the 54 and the 36 would be jealous

Thus, our GCF must be 2*3*3, or 18.