Factors and Multiples — Answer Key
Part A: Multiple Choice
Circle the best answer for each question.
1. Red lights flash every 6 seconds and blue lights every 8 seconds. They flash together and next flash together after how many seconds?
A) 14
B) 24
C) 36
D) 48
Find the LCM of 6 and 8: multiples of 8 are 8, 16, 24... and 24 is also a multiple of 6. They next flash together at 24 seconds.
2. How many composite numbers are between 1 and 15?
A) 6
B) 7
C) 8
D) 9
The composite numbers from 1 to 15 are 4, 6, 8, 9, 10, 12, 14, and 15. That gives 8 composite numbers in all.
3. What is the GCF of 45 and 75?
A) 5
B) 9
C) 15
D) 25
15 is the largest number that divides both 45 and 75 evenly: 45 ÷ 15 = 3 and 75 ÷ 15 = 5.
4. What is the LCM of 14 and 21?
A) 28
B) 42
C) 84
D) 294
Multiples of 21 are 21, 42... and multiples of 14 are 14, 28, 42... The smallest they share is 42.
Part B: Fill in the Blank
Write the correct answer on each line.
1. The prime factorization of 84 is 2 × 2 × 3 × 7.
84 = 4 × 21 = (2 × 2) × (3 × 7). So 84 = 2 × 2 × 3 × 7, and the missing prime factor is 7.
2. The LCM of 10 and 15 is 30.
Multiples of 15 are 15, 30... and multiples of 10 are 10, 20, 30... The smallest they share is 30.
3. A number divisible by 9 is also always divisible by 3.
9 = 3 × 3, so any number divisible by 9 already contains 3 as a factor and is divisible by 3 too.
4. The GCF of 56 and 84 is 28.
28 is the largest number that divides both 56 and 84 evenly: 56 ÷ 28 = 2 and 84 ÷ 28 = 3.
5. The smallest number divisible by both 8 and 12 is 24.
This is asking for the LCM. Multiples of 12 are 12, 24... and 24 ÷ 8 = 3 with no remainder, so 24 works.