The Number System Least Common Multiples 1 © 2013 Meredith S. Moody.

The Number System

Least Common Multiples

1© 2013 Meredith S. Moody

Objective: You will be able to…

Find multiples for a given number

Find common multiples for a pair or set of numbers

Find the first (lowest) common multiple for a pair or set of numbers


Vocabulary Common multiple: A number

exactly divisible by each number in a given set

Least common multiple: The lowest multiple exactly divisible by each number in a given set


Guided practice 1, multiples of single numbers

Find the multiples of 10 up to 100• 10, 20, 30, 40, 50, 60, 70, 80, 90, & 100• Find multiples by using multiplication: 10

x 1, 10 x 2... Find the multiples of 5 up to 50

• 5, 10, 15, 20, 25, 30, 35, 40, 45, 50• Find by using multiplication: 5 x 1, 5 x 2…

Find the multiples of 7 up to 100• 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98• Find by using multiplication: 7 x 1, 7 x 2…


Example 1, finding least common multiples, ‘chart’ method

Find the least common multiple of 2 and 3

Continue listing multiples until there is a common multiple

2 2 x 1 = 2 2 x 2 = 4 2 x 3 = 6

3 3 x 1 = 3 3 x 2 = 6 3 x 3 = 9


Finding least common multiples, ‘prime factorization’ method

We can use prime factorization to find the least common multiple for a set of numbers

Break down each number in the set into its prime factors

For each prime factor, use the highest frequency, then find the product of the primes


Example 1, finding least common multiples, ‘prime factorization’ method

Use prime factorization to find the least common multiple for 8 and 12

The prime factorization of 8 is 2 x 2 x 2 The prime factorization of 12 is 2 x 2 x 3 The prime factor 2 appears three times in the

factorization of 8 (most frequent) The prime factor 3 appears one time in the

factorization of 12 (most frequent) Multiply 2 x 2 x 2 x 3 = 24 The least common multiple of 8 and 12 is 24


Guided practice 1, least common multiple

Find the least common multiple for 6 and 8• These numbers are small, so using a chart makes

the most sense

• The least common multiple for 6 & 8 is 24

8

6 x 1 = 6 6 x 2 = 12 6 x 3 = 18 6 x 4 = 24

8 x 1 = 8 8 x 2 = 16 8 x 3 = 24 8 x 4 = 32

© 2013 Meredith S. Moody


Find the least common multiple for 12 and 30• These numbers are medium-sized; either method works

• OR, the prime factorization of 12 is 2 x 2 x 3; the prime factorization of 30 is 2 x 3 x 5

• The prime factor 2 occurs once in 30 but twice in 12, so twice is the highest frequency

• The prime factor 3 occurs once in both numbers, so once is the highest frequency

• The prime factor 5 occurs once in 30, so once is the highest frequency

• 2 x 2 x 3 x 5 = 60

9

12 x 1 = 12 12 x 2 = 24 12 x 3 = 36 12 x 4 = 48 12 x 5 = 60

30 x 1 = 30 30 x 2 = 60 30 x 3 = 90 30 x 4 = 120 30 x 5 = 150

© 2013 Meredith S. Moody


Find the least common multiple for 25 and 75• These numbers are large, so using the prime

factorization method makes the most sense • The prime factorization of 25 is 5 x 5• The prime factorization of 75 is 3 x 5 x 5• The prime factor 3 occurs once in 75; once is the

highest frequency• The prime factor 5 occurs twice in both 25 and 75;

twice is the highest frequency• 3 x 5 x 5 = 75• 75 is the least common multiple for 25 and 75



A local restaurant is offering a free meal to every 25th customer and a free hat to every 12th customer. Which customer will be the first to get both a free meal and a free hat?

Find the least common multiple Prime factorization of 25 is 5 x 5 Prime factorization of 12 is 2 x 2 x 3 2 x 2 x 3 x 5 x 5 = 300 The 300th customer will get both items


Common mistakes One mistake people make when finding the

least common multiple is to simply multiply the two numbers together

This DOES give A common multiple, but it might not always be the LEAST common multiple

People usually do this because they are lazy!...or don’t know any better


Common mistakes Another common mistake is that people find the

greatest common factor INSTEAD of the least common multiple and vice versa

How do you remember what to do? One way to remember is that greatest common factor

has “factor” in it – this sounds like “fraction” and fractions are smaller than whole numbers factors are smaller than the original numbers (at least one of them)

Least common multiple has the word “multiple” in it – this sounds like “multiply” and when we multiply we get larger numbers common multiples are larger than the original numbers (at least one of them)

© 2013 Meredith S. Moody 13

Least common multiple: You try

Find the least common multiple of the following sets of numbers

4 & 10• 20

12, 15, & 20• 60

4 & 18• 36


The Number System Least Common Multiples 1 © 2013 Meredith S. Moody.

Documents

Transcript of The Number System Least Common Multiples 1 © 2013 Meredith S. Moody.