math 35 fall term professor carl scarbnick [email protected] topic: least common multiples

Math 35 Fall TermProfessor Carl Scarbnick

[email protected]

Topic: Least Common Multiples

Section 1.6 Least Common Multiples

Objectives

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1. Find the least common multiple (LCM)

2. Find the LCM using multiples of the largest number

3. Find the LCM using prime factorization

(Warning: The graphics in these slides depend on the screen

resolution of your monitor. Please use the handout passed

out on Aug 27, 2007 if you experience any problems.)

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1.6 Least Common Multiples

Objective 1. Find the least common multiples

The least common multiple (LCM) of two whole numbers is the smallest number divisible by both these numbers.

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Objective 1. Find the least common multiples

Example:

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63

The smallest number in each list is 28. This means that 28 is the least common multiple (LCM) of 4 and 7.

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Objective 2. Find the LCM using multiples of

the largest number

When the numbers are small, we can find the LCM with the following procedure

Write out the multiples of the larger number and find the smallest one that is a multiple of the smaller number.

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Objective 2. Find the LCM using multiples of

the largest number

Example: Find the least common multiple of 5 and 6.

The multiples of 6 are 6, 12, 18, 24, 30, 36, …

The smallest multiple where the last digit is zero or five is the number, 30. Therefore, 30 is the least common multiple of 5 and 6.

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Objective 3. Find the LCM using

prime factorization.

Example: The prime factorization of 6 = 2 x 3. The prime factorization of 9 = 3 x 3.

Notice that the prime factorization of 6 and the prime factorization of 9 includes the number, 3.

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Example: The prime factorization of 6 = 2 x 3. The prime factorization of 9 = 3 x 3.

The LCM of 6 and 9 is 2 x 3 x 3 = 18.

Factors of 6

Factors of 9

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Example: The LCM of 6 and 9 is 2 x 3 x 3 = 18. We do not repeat the prime factors shared by the numbers, 6 and 9.

Note that 6 x 9 = (2 x 3) x (3 x 3) = 54 is a multiple of 6 and 9. However, it is not the least common multiple, since it repeats a prime factor shared by both numbers.

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General Procedure for computing the LCM with prime factorization:

Step 1: Write the prime factorization of each number.

Step 2: Take the product of the prime numbers that appear in each list. If a prime number appears more than one time in a list, use it the most number of times it appears in a list.

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Problem 1: Find the least common multiple of 14 and 36

Step 1: The prime factorization of 14 is 14 = 2 x 7 The prime factorization of 36 is 36 = 2 x 2 x 3 x 3

Step 2

LCM = 2 x 2x 3 x 3 x 7 = 252

Prime Numbers 2 3 7

Most number of times in a list

2 2 1

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Problem 2: Find the least common multiple of 24 and 62.

Step 1: The prime factorization of 24 is 24 = 2 x 2 x 2 x 3 The prime factorization of 62 is 62 = 2 x 31

Step 2

LCM = 2 x 2 x 2 x 3 x 31 = 744

Prime Numbers 2 3 31

Most number of times in a list

3 1 1

math 35 fall term professor carl scarbnick [email protected] topic: least common multiples

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