Least Common Multiple (LCM)

Show your work for each problem.

1.
What is the LCM of and ?
1. a)6
2. b)5
3. c)3
4. d)1
Answer: 6
Multiples of : . Multiples of : . The first common multiple is , so .
2.
Find the LCM of and .
Answer: 20
Since and have no common factors (they are coprime), .
3.
Find the LCM of and by listing multiples.
Answer: 6
1. List the first few multiples of 3 3, 6, 9, 12
2. List the first few multiples of 6 6, 12, 18
3. What is the smallest number that appears in both lists? 6
4.
Which statement about LCM is TRUE?
1. a)LCM is always smaller than both numbers
2. b)LCM is the same as GCF
3. c)LCM is always the product of two numbers
4. d)LCM is the smallest common multiple
Answer: LCM is the smallest common multiple
LCM stands for Least Common Multiple - the smallest number that is a multiple of all given numbers. It is NOT the same as GCF, NOT always the product, and is always greater than or equal to the largest number.
5.
Find the LCM of and .
Answer: 24
Multiples of : . Multiples of : . The first common multiple is , so .
6.
Use prime factorization to find the LCM of and .
Answer: 24
1. What is the prime factorization of 8? 2^3
2. What is the prime factorization of 12? 2^2 x 3
3. What is the highest power of 2 needed? 2^3
4. What is the highest power of 3 needed? 3
5. Multiply the highest powers: 2^3 x 3 = ? 24
7.
If , what must be true?
1. a)a divides b evenly
2. b)a and b are coprime
3. c)b divides a evenly
4. d)a equals b
Answer: a divides b evenly
If LCM(a, b) = b, then b is already a multiple of a. This means a divides b evenly. Example: LCM(3, 6) = 6 because 3 divides 6.
8.
Bus A arrives every minutes and Bus B arrives every minutes. If both arrive now, when will they arrive together again?
Answer: 30
1. What are the first few multiples of 10? 10, 20, 30, 40
2. What are the first few multiples of 15? 15, 30, 45
3. What is the smallest time when both buses arrive? 30
9.
Find the LCM of and .
Answer: 36
and . LCM = .
10.
Find the LCM of , , and .
Answer: 60
1. What is the prime factorization of 4? 2^2
2. What is the prime factorization of 6? 2 x 3
3. What is the prime factorization of 10? 2 x 5
4. What primes are needed? 2, 3, 5
5. Calculate 2^2 x 3 x 5 = ? 60
11.
Find the LCM of and .
Answer: 60
and . LCM = .
12.
What is the LCM of , , and ?
1. a)180
2. b)240
3. c)120
4. d)60
Answer: 120
, , . LCM = .
13.
To add , what is the LCD (least common denominator)?
Answer: 24
1. What are the denominators? 6 and 8
2. What is the prime factorization of 6? 2 x 3
3. What is the prime factorization of 8? 2^3
4. Calculate LCM(6, 8) = 2^3 x 3 = ? 24
14.
If and , what is ?
Answer: 24
Using : , so .
15.
Two lights flash. One flashes every seconds, the other every seconds. If they flash together now, when will they flash together again?
1. a)54 seconds
2. b)36 seconds
3. c)15 seconds
4. d)18 seconds
Answer: 18 seconds
. Multiples of : . Multiples of : . They flash together in seconds.