1. 5 4 . Least Common Multiple In the Real World
L E S S O N
5 4 .
Least Common Multiple
In the Real World
Ferry Boats Two ferry boats leave a loading platform at the same time. One of the ferry boats returns to the loading platform every 25 minutes. The other returns every 30 minutes. In the next 300 minutes, when will they return at the same time?
You can use multiples to answer the question above. A multiple of a number is the product of the number and any nonzero whole number.
Multiples of 2: 2, 4, 6, 8,
10,
12,
14, …
The three dots show that the pattern continues forever.
A multiple shared by two or more numbers is a common multiple.

2. 5 4 . 1 Least Common Multiple Finding a Common Multiple
L E S S O N
5 4 .
EXAMPLE 1
Finding a Common Multiple
Ferry Boats Two ferry boats leave a loading platform at the same time. One of the ferry boats returns to the loading platform every 25 minutes. The other returns every 30 minutes. In the next 300 minutes, when will they return at the same time?
You can use common multiples to answer this question. Begin by writing the multiples of 25 and 30.
Multiples of 25:
25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, …
150
300
Multiples of 30:
30, 60, 90, 120, 150, 180, 210, 240, 270, 300, …
150
300
Now identify the common multiples through 300.
The ferry boats will return to the loading platform at the same time in 150 minutes and in 300 minutes.
ANSWER

3. Finding the Least Common Multiple (LCM)
L E S S O N
5 4 .
NOTE
BOOK
Finding the Least Common Multiple (LCM)
The least common multiple of two or more numbers is the smallest of the common multiples. Below are two methods to find the LCM.
Method 1:
Start listing the multiples of each number. Then find the smallest of the common multiples.
Method 2:
Write the prime factorizations of the numbers. Multiply together the prime factors, using each prime factor the greatest number of times it is a factor of any of the numbers.

4. 5 4 . 2 Least Common Multiple Finding the LCM
L E S S O N
5 4 .
EXAMPLE 2
Finding the LCM
Find the LCM of 9 and 12.
Multiples of 9:
9, 18, 27, 36, 45, 54, … 36
Multiples of 12:
12, 24, 36, 48, … 36
ANSWER
The LCM of 9 and 12 is 36.

5. 5 4 . 3 Least Common Multiple Using Prime Factorization
L E S S O N
5 4 .
EXAMPLE 3
Using Prime Factorization
Find the LCM of 42 and 60 using prime factorization. 1
Write the prime factorizations. Circle any common factors.
42 = 2  3  7
60 = 2  2  3  5 2
Multiply together the prime factors, using each circled factor the greatest number of times it occurs in either factorization.
2  2  3  5  7
= 420
ANSWER
The LCM of 42 and 60 is 420.