1. Warm Up
Write the prime factorization of each number.
1. 68
2. 225
3. 940
Find the greatest common factor.
4. 27 and 45
5. 32 and 80
6. 50 and 71
22 · 17
32 · 52
22 · 5 · 47 9 16 1

2. Learn to find the least common multiple of two or more whole numbers.

3. Insert Lesson Title Here
Vocabulary
multiple
least common multiple (LCM)

4. The maintenance schedule on Ken’s pickup truck shows that the tires should be rotated every 7,500 miles and that the oil filter should be replaced every 5,000 miles. What is the lowest mileage at which both services are due at the same time? To find the answer you can use least common multiples.

5. A multiple of a number is a product of that number and a whole number
A multiple of a number is a product of that number and a whole number. Some multiples of 7,500 and 5,000 are as follows:
7,500: 7,500, 15,000, 22,500, 30,000, 37,500, 45,000, . . .
5,000: 5,000, 10,000, 15,000, 20,000, 25,000, 30,000, . . .
A common multiple of two or more numbers
is a number that is a multiple of each of the
given numbers. So 15,000 and 30,000 are
common multiples of 7,500 and 5,000.

6. The least common multiple (LCM) of two or more numbers is the common multiple with the least value. The LCM of 7,500 and 5,000 is 15,000. This is the lowest mileage at which both services are due at the same time.

7. Example : Using a List to Find the LCM
Find the least common multiple (LCM).
A. 2, 7
2: 2, 4, 6, 8, 10, 12, 14
List some multiples of each
number.
7: 7, 14, 21, 28, 35
Find the least value that is
in both lists.
The LCM is 14.
B. 3, 6, 9
3: 3, 6, 9, 12, 15, 18, 21
List some multiples of each
number.
6: 6, 12, 18, 24, 30
9: 9, 18, 27, 36, 45
Find the least value that is
in all the lists.
The LCM is 18.

8. Insert Lesson Title Here
Try This: Example
Find the least common multiple (LCM).
A. 3, 7
List some multiples of each
number.
3: 3, 6, 9, 12, 15, 18, 21
7: 7, 14, 21, 28
Find the least value that is
in both lists.
The LCM is 21.
B. 2, 6, 4
2: 2, 4, 6, 8, 10, 12, 14
List some multiples of each
number.
6: 6, 12, 18
4: 4, 8, 12
Find the least value that is
in all the lists.
The LCM is 12.

9. Example : Using Prime Factorization to Find the LCM
Find the least common multiple (LCM).
A. 60, 130
60 = 2 · 2 · 3 · 5
Write the prime factorization of
each number.
130 = 2 · 5 · 13
Circle the common prime factors.
2, 2, 3, 5, 13
List the prime factors, using
the circled factors only once.
2 · 2 · 3 · 5 · 13
Multiply the factors in the list.
The LCM is 780.

10. Example : Using Prime Factorization to Find the LCM
Find the least common multiple (LCM).
B. 14, 35, 49
14 = 2 · 7
Write the prime factorization of
each number.
35 = 5 · 7
Circle the common prime factors.
49 = 7 · 7
List the prime factors, using
the circled factors only once.
2, 5, 7, 7
2 · 5 · 7 · 7
Multiply the factors in the list.
The LCM is 490.

11. Insert Lesson Title Here
Try This: Example 2A
Find the least common multiple (LCM).
A. 50, 130
50 = 2 · 5 · 5
Write the prime factorization of
each number.
110 = 2 · 5 · 11
Circle the common prime factors.
List the prime factors, using
the circled factors only once.
2, 5, 5, 11
2 · 5 · 5 · 11
Multiply the factors in the list.
The LCM is 550.

12. Insert Lesson Title Here
Try This: Example
B. 18, 36, 54
18 = 2 · 3 · 3
Write the prime factorization of
each number.
36 = 2 · 2 · 3 · 3
Circle the common prime factors.
54 = 2 · 3 · 3 · 3
List the prime factors, using
the circled factors only once.
2, 2, 3, 3, 3
2 · 2 · 3 · 3 · 3
Multiply the factors in the list.
The LCM is 108.

13. Additional Example : Application
Mr. Washington will set up the band chairs all in rows of 6 or all in rows of 8. What is the least number of chairs he will set up?
Find the LCM of 6 and 8.
6 = 2 · 3
8 = 2 · 2 · 2
The LCM is 2 · 2 · 2 · 3 = 24.
He will set up at least 24 chairs.

14. Insert Lesson Title Here
Try This: Example
Two satellites are put into orbit over the same location at the same time. One orbits the earth every 24 hours, while the second completes an orbit every 18 hours. How much time will elapse before they are once again over the same location at the same time?
Find the LCM of 24 and 18.
24 = 2 · 2 · 2 · 3
18 = 2 · 3 · 3
The LCM is 2 · 2 · 2 · 3 · 3 = 72.