1. EQ: How can we find the least common multiple of two numbers?
Lesson 2.2 -Together again: Investigating Multiples and Least Common Multiples
EQ: How can we find the least common multiple of two numbers?

2. Common Multiple
A common multiple is a number that is a multiple of two or more numbers.
It is like counting by a number.
Example:
3, 6, 9, 12,15 18, 21,24, 27, 30.
6, 12, 18, 24, 30, 36, 42, 48, 54, 60.

3. Question
Many things in life repeat over and over again in a regular cycle.
A news radio station in Atlanta broadcasts a weather report every 12 minutes.
A ferry crosses Lake Champlain from Vermont to New York every 36 minutes.
Big Ben in London, England, chimes every 15 minutes.
At 8:00 am, the radio station broadcasted a weather report at the same time that Big Ben chimed.
In how many minutes will the radio station broadcast a weather report and Big Ben chime at the same time again? List multiples of each and explain your reasoning.

4. Question
Many things in life repeat over and over again in a regular cycle.
A news radio station in Atlanta broadcasts a weather report every 12 minutes.
A ferry crosses Lake Champlain from Vermont to New York every 36 minutes.
Big Ben in London, England, chimes every 15 minutes.
At 8:00 am, the radio station broadcasted a weather report at the same time that Big Ben chimed.
In how many minutes will the radio station broadcast a weather report and Big Ben chime at the same time again? List multiples of each and explain your reasoning.
The multiples of 12 are 12, 24, 36, 48, 60, . . .
The multiples of 15 are 15, 30, 45, 60, . . .
The radio station will broadcast a weather report and Big Ben will chime at the
same time in 60 minutes.

5. Least Common Multiple (LCM)
The least common multiple, abbreviated as LCM, is the smallest multiple (other than zero) that two or more numbers have in common.
One method, shown earlier is to count by that number.
A more efficient method to determine the least common multiple of two numbers is to use prime factors. Let’s consider using prime factorization to determine the LCM of 36 and 15.

6. Here is How
To calculate the LCM of 36 and 15, first write the prime factors of each, and identify the unique prime factors.

7. Here is How Next, determine the greatest power of each prime factor.
Finally, multiply those factors.
LCM = 22 x 32 x 5
= 4 x 9 x 5
= 180
The LCM of 36 and 15 is 180.

8. Your Turn
Determine the least common multiple for each pair of numbers using prime factorization. Create factor trees if you need help determining the prime factorization of each number.
a. 8 and b. 126 and 45
c. 165 and d. 6, 7, and 45

9. Your Turn
Determine the least common multiple for each pair of numbers using prime factorization. Create factor trees if you need help determining the prime factorization of each number.
a. 8 and b. 126 and 45
8 = = 2 x 32 x 7
12 = 22 x = 32 x 5
LCM = 23 x 3 = LCM = 2 x 32 x 5 x 7 = 630
c. 165 and d. 6, 7, and 45
165 = 3 x 5 x = 2 x 3
22 5= 2 x = 1 x 7
LCM = 2 x 3 x 5 x 11 = = 32 x 5
LCM = 2 x 32 x 5 x 7 = 630