Prime Factors Worksheet Grade 3-08 - Free Printable

Let's solve each question step by step and explain the reasoning behind the correct answers.

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1. $2^3 \times 7$ is the prime factorization of which number?

We calculate:
$$
2^3 = 8, \quad 8 \times 7 = 56
$$

So, the number is 56.

✔ Answer: b. 56

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2. What is the prime factorization of 110?

Break down 110:
- 110 ÷ 2 = 55 → so 2 is a factor
- 55 ÷ 5 = 11 → so 5 is a factor
- 11 is a prime number

So: $110 = 2 \times 5 \times 11$

✔ Answer: d. $2 \times 5 \times 11$

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3. What is the prime factorization of 99?

Break down 99:
- 99 ÷ 3 = 33 → 3 is a factor
- 33 ÷ 3 = 11 → another 3
- 11 is prime

So: $99 = 3 \times 3 \times 11 = 3^2 \times 11$

Now check options:
- a. $3^2 \times 11$ → ✔ Correct
- b. $3^3 \times 11 = 27 \times 11 = 297$ → wrong
- c. $9 \times 22$: 9 and 22 are not primes → invalid
- d. "all of these" → false

✔ Answer: a. $3^2 \times 11$

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4. What is the prime factorization of 93?

Break down 93:
- 93 ÷ 3 = 31 → 3 is a factor
- 31 is a prime number

So: $93 = 3 \times 31$

Check options:
- a. $3 \times 30$: 30 is not prime → invalid
- b. $3 \times 31$ → ✔ Correct
- c. $3 \times 3 \times 3 \times 3 = 81$ → wrong
- d. 0 → nonsense

✔ Answer: b. $3 \times 31$

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5. The prime factors of 70 are

Break down 70:
- 70 ÷ 2 = 35 → 2 is a factor
- 35 ÷ 5 = 7 → 5 is a factor
- 7 is prime

So: $70 = 2 \times 5 \times 7$

✔ Answer: d. $2 \times 5 \times 7$

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6. The prime factors of 440 are

Break down 440:
- 440 ÷ 2 = 220
- 220 ÷ 2 = 110
- 110 ÷ 2 = 55
- 55 ÷ 5 = 11
- 11 is prime

So: $440 = 2 \times 2 \times 2 \times 5 \times 11 = 2^3 \times 5 \times 11$

Now check options:
- a. $2 \times 5 \times 7 = 70$ → no
- b. $2 \times 3 \times 5 \times 5 = 150$ → no
- c. $2 \times 2 \times 2 \times 5 \times 11 = 440$ → ✔ Correct
- d. $2 \times 3 \times 3 \times 5 \times 5 = 450$ → no

✔ Answer: c. $2 \times 2 \times 2 \times 5 \times 11$

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7. The prime factors of 40 are

Break down 40:
- 40 ÷ 2 = 20
- 20 ÷ 2 = 10
- 10 ÷ 2 = 5
- 5 is prime

So: $40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5$

Check options:
- a. $2 \times 2 \times 3 \times 5 \times 5 = 300$ → no
- b. $2 \times 3 \times 5 \times 5 = 150$ → no
- c. $2 \times 2 \times 2 \times 5 = 40$ → ✔ Correct
- d. $2 \times 3 \times 5 = 30$ → no

✔ Answer: c. $2 \times 2 \times 2 \times 5$

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8. ________ is the prime factorization of 24.

Break down 24:
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 is prime

So: $24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3$

Check options:
- a. $2 \times 2 \times 2 \times 3 \times 3 = 72$ → no
- b. $2 \times 2 \times 2 \times 3 = 24$ → ✔ Correct
- c. $2 \times 2 \times 2 \times 2 \times 3 = 48$ → no
- d. $2 \times 2 \times 3 \times 3 = 36$ → no

✔ Answer: b. $2 \times 2 \times 2 \times 3$

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9. Which of the following contains all prime numbers?

Recall: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

Check each option:

- a. 2, 3, 5, 7, 11 → All are primes ✔
- b. 2, 3, 5, 7, 9 → 9 = 3×3 → not prime ✘
- c. 2, 4, 5, 7, 11 → 4 = 2×2 → not prime ✘
- d. 2, 3, 5, 7, 12 → 12 = 2×2×3 → not prime ✘

✔ Answer: a. 2, 3, 5, 7, 11

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10. The prime factor of 15 is

Break down 15:
- 15 ÷ 3 = 5 → both 3 and 5 are prime

So: $15 = 3 \times 5$

Check options:
- a. $3 \times 5$ → ✔ Correct
- b. $15 \times 1$: 15 is not prime, and 1 is not considered a prime → invalid
- c. $3 \times 5 \times 5 = 75$ → too big
- d. $3 \times 5 \times 2 = 30$ → too big

✔ Answer: a. $3 \times 5$

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✔ Final Answers:

1. b. 56
2. d. $2 \times 5 \times 11$
3. a. $3^2 \times 11$
4. b. $3 \times 31$
5. d. $2 \times 5 \times 7$
6. c. $2 \times 2 \times 2 \times 5 \times 11$
7. c. $2 \times 2 \times 2 \times 5$
8. b. $2 \times 2 \times 2 \times 3$
9. a. 2, 3, 5, 7, 11
10. a. $3 \times 5$

Let me know if you'd like this as a printable version or with explanations in simpler terms!