Bell Work: Solve for x: 5y + x – 2y – 4 + 3x = 0

Answer: x = -3/4 y + 1

Lesson 43: Least Common Multiple, Least Common Multiples of Algebraic Expressions

If we are given the numbers 4, 5, and 8 And are asked to find the smallest number that is evenly divisible by each of the numbers, a reasonable guess would be the product of the numbers, which is 160, because we know that each of the numbers will divide 160 evenly 160/4 = /5 = 32160/8 = 20

But 160 is not the smallest number that is evenly divisible by the three numbers. The number 40 is. 40/4 = 10 40/5 = 8 40/8 = 5

We call the smallest number that can be divided evenly by each of a group of specified numbers the least common multiple (LCM) of the specified numbers.

We can find the LCM of some numbers by making mental calculations, but it is nice to have a special procedure to use if some of the numbers are large numbers. The procedure is as follows:

1. Write each number as a product of prime factors. 2. Compute the LCM by using every factor of the given numbers as a factor of the LCM. Use each factor the greatest number of times it is a factor in any of the numbers.

To demonstrate this procedure we will find the LCM of 18, 81, and 500 First we write each number as a product of prime factors: 18 = 2 x 3 x 3 81 = 3 x 3 x 3 x = 2 x 2 x 5 x 5 x 5

Now we find the LCM by using the procedure in Step 2. The number 2 is a factor of both 18 and 500. it appears twice in 500, so it will appear twice in the LCM. 2 x 2

The number 3 is a factor of both 18 and 81. it appears four times in 81, so it will appear four times in the LCM. 2 x 2 x 3 x 3 x 3 x 3

Therefore, 40,500 is the smallest number that is evenly divisible by each of the three numbers 18, 81, and 500.

Example: Find the LCM of 8, 15, and 100.

Answer: 8 = 2 x 2 x 2 15 = 3 x = 2 x 2 x 5 x 5 2 x 2 x 2 x 3 x 5 x 5 = 600

Example: Find the Least Common Multiple of 30, 75, and 80.

Answer: 30 = 2 x 3 x 5 75 = 3 x 5 x 5 80 = 2 x 2 x 2 x 2 x 5 2 x 2 x 2 x 2 x 3 x 5 x 5 = 1200

Practice: Find the LCM of 560, 588, and 1250.

Answer: 560 = 2 x 2 x 2 x 2 x 5 x = 2 x 2 x 3 x 7 x = 2 x 5 x 5 x 5 x 5 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 5 x 7 x 7 = 1, 470, 000

The least common multiple is most often encountered when it is used as the least common denominator. If we are asked to add the fractions ¼ + 5/8 + 7/12 We rewrite each of these fractions as a fraction whose denominator is 24, which is the least common multiple of 4, 8 and 12. 6/ / /24 = 35/24

In lesson 44 we will discuss the method of adding algebraic fractions. To prepare for that, we will practice finding the least common multiple of algebraic expressions.

Example: Find the least common multiple of 15a b and 10ab. 23

Answer: 15a b = 3 x 5 x a x a x b 10ab = 2 x 5 x a x b x b x b LCM = 2 x 3 x 5 x a x a x b x b x b = 30a b

Practice: Find the LCM of 4x m and 6x m. 23

Answer: 4x m = 2  2  x  x  m 6x m = 2  3  x  x  x  m LCM = 2  2  3  x  x  x  m = 12x m 2 3 3

Practice: Find the LCM of 12x am and 14x am

Answer: 84x am 34

HW: Lesson 43 #1-30