Basic Math and Pre-Algebra

PART 1. The World of Numbers

CHAPTER 4. Factors and Multiples

Least Common Multiple

If you attend a baseball game at which every twelfth person admitted to the stadium receives a free key ring and every twentieth person gets a free t-shirt, where should you be in line if you want both? You want a spot that’s a multiple of 12, so that you get the key ring, but you want it to also be a multiple of 20, so you snag a t-shirt. You also don’t want to be too far back in line, so the smallest number that does both jobs is what you’re after. Many people would look to be #240 in line because that’s 12 x 20, but you can move, if not to the head of the line, at least farther up.

The least common multiple, or LCM, of two (or more) numbers is the smallest number that is a multiple of both. If the LCM is a multiple of both numbers, then both numbers divide the LCM, and both are factors of the LCM.

DEFINITION

The least common multiple of two or more numbers is the smallest number that has each of the numbers as a factor.

The least common multiple of 12 and 20 is much smaller than 240. 240 is a multiple of 12 and of 20. It is a common multiple, but it’s not the least, or smallest, common multiple. If you look at lists of the multiples of 12 and 20, you’ll see that 60 is the smallest number to show up in both lists.

Multiples of 12: 12, 24, 36, 48, 60, 72 ...

Multiples of 20: 20, 40, 60, 80 ...

You can stake out the 60th spot in line at the game, get a good seat, and go home with the key ring and the t-shirt.

Making a list of the multiples of each number and looking for the first number to pop up on both lists is actually a workable method of finding the least common multiple, if the numbers aren’t too big. For large numbers, however, it could take a very long time and a lot of multiplying.

Instead, you’ll want to start by finding the prime factorization of both numbers. Let’s start with an example with small numbers and then try another with larger numbers.

Suppose you need to find the least common multiple of 12 and 15. Find the prime factorization of each number.

12 = 2 x 2 x 3

15 = 3 x 5

Identify any common factors. In this example, 12 and 15 both have a factor of 3. Build the LCM by starting with the common factor or factors, then multiplying by the other factors of 12 and 15 that are not common. 12 and 15 have 3 in common, so that goes into the LCM just once, and then you collect the other factors of 12, which are 2 x 2, and the other factor of 15, the 5.

LCM = 3 x 2 x 2 x 5

The 3 is the common factor, the two 2s come from the factorization of 12, and the 5 is from the 15. Multiply to find that the LCM is 60. 60 is 12 x 5 and 15 x 4, and it’s the smallest multiple 12 and 15 share.

Ready to find the least common multiple of 98 and 168? Start by finding the prime factorization of each number. You might want to use a factor tree.

98 = 2 x 7 x 7

168 = 2 x 2 x 2 x 3 x 7

The two numbers have a 2 and a 7 in common, so start building the LCM with those.

LCM = 2 x 7 x ?

You need another 7 from the 98 and another two 2s and a 3 from the 168.

LCM = 2 x 7 x 7 x 2 x 2 x 3 = 1,176

The multiplication for that one might be challenging, but building lists of multiples up into the thousands wouldn’t have been an efficient method.

WORLDLY WISDOM

If two numbers are relatively prime, their product will be their LCM.

CHECK POINT

Find the least common multiple of each pair of numbers.

21. 14 and 35

22. 45 and 105

23. 286 and 715

24. 21 and 20

25. 88 and 66

The Least You Need to Know

• A prime number has no factors other than itself and one.

• A number that is not prime is composite, except 1, which is in a category all its own.

• The prime factorization of a number writes the number as a product of primes.

• The greatest common factor of two numbers is the largest number that is a factor of both.

• The least common multiple of two numbers is the smallest number that is a multiple of both.