Finding the Least Common Multiples


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The multiples of a number are the products of that number andthe whole numbers. For instance the multiples of 5 are thefollowing.

A number that is a multiple of two or more numbers is called acommon multiple of these numbers. To find the common multiples of6 and 8, we first list the multiples of 6 and the multiples of 8separately.

So the common multiples of 6 and 8 are 0, 24, 48, ... . Of thenonzero common multiples, the least common multiple of 6 and 8 is24.

Definition

The least common multiple (LCM) of two ormore numbers is the smallest nonzero number that is a multiple ofeach number.

A shortcut for finding the LCM—faster than listingmultiples—involves prime factorization.

To Compute the Least Common Multiple (LCM)

find the prime factorization of each number,
identify the prime factors that appear in each factorization, and
multiply these prime factors, using each factor the greatest number of times that itoccurs in any of the factorizations.

EXAMPLE 1

Find the LCM of 8 and 12.

Solution

We first find the prime factorization of each number.

The factor 2 appears three times in the factorization of 8 andtwice in the factorization of 12, so it must be included threetimes in forming the least common multiple.

As always, it is a good idea to check that our answer makessense. We do so by verifying that 8 and 12 really are factors of24.

EXAMPLE 2

Find the LCM of 5 and 9.

Solution

First we write each number as the product of primes.

To find the LCM we multiply the highest power of each prime.

So the LCM of 5 and 9 is 45. Note that 45 is also the productof 5 and 9. Checking our answer, we see that 45 is a multiple ofboth 5 and 9.

TIP

If two or more numbers have no common factor (other than 1),the LCM is their product.

Now let's find the LCM of three numbers.

EXAMPLE 3

Find the LCM of 3, 5, and 6.

Solution

First we find the prime factorizations of these three numbers.

3 = 3

5 = 5

6 = 2 × 3

The LCM is therefore the product 2 × 3 × 5, which is 30.Note that 30 is a multiple of 3, 5, and 6, which supports ouranswer.

EXAMPLE 4

A gym that is open every day of the week offers aerobicclasses everythird day and gymnastic classes every fourth day.You took both classesthis morning. In how many days will the gymoffer both classes on thesame day?

Solution

To answer this question, we ask: What is the LCM of 3 and 4?As usual, we begin by finding prime factorizations.

To find the LCM, we multiply 3 by .

Both classes will be offered again on the same day in 12 days.