5.5Least Common Multiple

Find the Least Common Multiple (LCM) by Listing

Least Common Multiple (LCM): The smallest natural number that is divisible by all the given numbers.

To find the LCM by listing, list the multiples of the greatest given number until you find a multiple that is divisible by all the other given numbers.

Example: Find the LCM of 4 and 6 6: 6, 12, 18, 24, 30, 36, 42, 48, … 4: 4, 8, 12, 16, 20, 24, 28, …

Find the LCM of 36 and 120

120: 120, 240, 360, …

36 does not divide 120 evenly, so we go to the next multiple of 120

36 does not divide 240 evenly, so we go to the next multiple of 120

36 divides 360 evenly 10 times. This is the LCM

LCM of 36 and 120 is 360

Find the LCM using Prime Factorization

1. Find the prime factorization of each given number.

2. Write a factorization that contains each prime factor the greatest number of times it occurs in the factorization. Or, if you prefer to use exponents, the factorization contains each prime factor raised to the greatest exponent that occurs in the factorization.

3. Multiply to get the LCM.

Find the LCM of a Set of Monomials

Same procedure as finding LCM for a set of numbers.

When finding the LCM for a group of monomials include each coefficient and variable to the highest degree.

Let’s look at some examples

Find the LCM of 24, 90, and 70 using prime factorization.

24 = 23 • 3 90 = 2 • 32 • 5 70 = 2 • 5 • 7LCM (24, 90, 70) = 23 • 32 • 5 • 7 = 8 • 9 • 5 • 7 = 2520

23 has the greatest exponent for 2,

32 has the greatest exponent for 3,

5 has the greatest exponent for 5,

and 7 has the greatest exponent for 7.

Find the LCM of 18x3y and 24xz2.

18x3y = 2 • 32 • x3 • y 24xz2 = 23 • 3 • x • z2

LCM(18x3y, 24xz2) = 23 • x3 • y • z2

= 72x3yz2

23 has the greatest exponent for 2,32 has the greatest exponent for 3,x3 has the greatest exponent for x,y has the greatest exponent for y,

and z2 has the greatest exponent for z.

You Try… Find the LCM

18 and 24

4, 6, and 9

9m2n and 12m4

Write Factions as equivalent Fractions with the Least

Common Denominator (LCD)

Least Common Denominator (LCD): The least common multiple of the denominator

€

Write 7

12 and

2

15 as equivalent fractions with the least common denominator

1st, find the LCM of the denominator

15: 15, 30, 45, 60, 75, 90, ….12: 12, 24, 36, 48, 60, 72, 84, …

Both denominators need to be 60:

€

7

12=

7 ⋅5 = 35

12 ⋅5 = 60=

35

602

15=

2 ⋅4 = 8

15 ⋅4 = 60=

8

6035

60,

8

60

€

Write 3

8x and

5

6x2 as equivalent rational expressions with the LCD.

We know that the LCD for the two fractions will be the LCD between 8 and 6 with an x2 involved.

8: 8, 16, 24, 32, 40, 48, …6: 6, 12, 18, 24, 30, 36, ….

So the LCD will be 24x2

€

3

8x=

3⋅3x = 9x

8x ⋅3x = 24x 2=

9x

24x 2

5

6x 2=

5 ⋅4 = 20

6x 2 ⋅4 = 24x 2=

20

24x 2