Prime Factorization

Show your work for each problem.

1.
What is the prime factorization of 12?
1. a)
2. b)
3. c)
4. d)
Answer:
. The other options contain composite numbers (6 or 4) which must be factored further.
2.
Which number is NOT a prime factor of 30?
1. a)3
2. b)6
3. c)2
4. d)5
Answer: 6
. The number 6 is not prime (it equals ), so it cannot be a prime factor.
3.
What is the smallest prime factor of 45?
Answer: 3
. Since 45 is odd, 2 is not a factor. The smallest prime factor is 3.
4.
How many prime factors does 8 have? (Count with repetition)
Answer: 3
. There are three 2s, so 8 has 3 prime factors (counting repetition).
5.
Find the prime factorization of 18.
Answer: 2 × 3²
1. Is 18 even? If yes, divide by 2. What do you get? 9
2. Is 9 prime or composite? composite
3. Factor 9 into two numbers. 3 * 3
4. Write the complete prime factorization. 2 * 3 * 3
6.
Find the prime factorization of 20.
Answer: 2² × 5
1. 20 is even. Divide by 2. What do you get? 10
2. 10 is also even. Divide by 2 again. What do you get? 5
3. Is 5 prime? yes
4. Write the prime factorization in exponential form. 2^2 * 5
7.
What is the prime factorization of 72?
1. a)
2. b)
3. c)
4. d)
Answer:
. Note that is not a prime factorization because 8 and 9 are composite.
8.
If , what is the value of ?
1. a)100
2. b)150
3. c)250
4. d)200
Answer: 200
9.
What is the largest prime factor of 84?
Answer: 7
. The prime factors are 2, 3, and 7. The largest is 7.
10.
How many distinct prime factors does 60 have?
Answer: 3
. The distinct prime factors are 2, 3, and 5. There are 3 distinct primes.
11.
Find the prime factorization of 48.
Answer: 2⁴ × 3
1. Divide 48 by 2. What do you get? 24
2. Divide 24 by 2. What do you get? 12
3. Divide 12 by 2. What do you get? 6
4. Divide 6 by 2. What do you get? 3
5. Write the complete prime factorization in exponential form. 2^4 * 3
12.
Find the prime factorization of 90.
Answer: 2 × 3² × 5
1. Is 90 divisible by 2? yes
2. What is 90 / 2? 45
3. Is 45 divisible by 3? If yes, what is 45 / 3? 15
4. What is 15 / 3? 5
5. Write the prime factorization in exponential form. 2 * 3^2 * 5
13.
What is the prime factorization of 225?
1. a)
2. b)
3. c)
4. d)
Answer:
. Note: equals 225 but is not a prime factorization since 15 is composite.
14.
If and , what is the value of ?
Answer: 5
. So and , giving .
15.
Find the prime factorization of 360.
Answer: 2³ × 3² × 5
1. 360 ends in 0, so it's divisible by 10. What is 360 / 10? 36
2. Factor 10 into primes. 2 * 5
3. Now factor 36 into primes. 2^2 * 3^2
4. Combine all prime factors: 10 x 36 = (2 x 5) x (2^2 x 3^2). 2^3 * 3^2 * 5
16.
Find the prime factorization of 504.
Answer: 2³ × 3² × 7
1. 504 is even. Divide by 2 until you get an odd number. What's the odd number? 63
2. How many times did you divide by 2? 3
3. Now factor 63. Is it divisible by 3? yes
4. What is 63 / 3 / 3? 7
5. Write the complete prime factorization. 2^3 * 3^2 * 7