Numbers and operations · Ages 12–13

Prime factorisation worksheet

Part of: Working with numbers

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Numbers and operations · Ages 12–13

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.Write 240 as a product of prime factors: 240 = 2^□ × 3^□ × 5^□
Fill in each blank
Answer:
2.The largest prime factor of 78 is .
Answer:
3.Write 300 as a product of prime factors: 300 = 2^□ × 3^□ × 5^□
Fill in each blank
Answer:
4.336 = 2^□ × 3^□ × 7^□
Fill in each blank
Answer:
5.Break 60 into prime factors by dividing again and again. Put the steps in order.
- A. 15 ÷ 3 = 5
- B. 60 ÷ 2 = 30
- C. 5 ÷ 5 = 1
- D. 30 ÷ 2 = 15
Write the letters in the right order
Answer:
6.138 = 2¹ × 3¹ × . What prime goes in ?
Answer:
7.Break 72 × 30 into prime factors: 72 × 30 = 2^□ × 3^□ × 5^□
Fill in each blank
Answer:
8.Break 60 × 9 into prime factors: 60 × 9 = 2^□ × 3^□ × 5^□
Fill in each blank
Answer:
9.Break 84 into prime factors by dividing again and again. Put the steps in order.
- A. 7 ÷ 7 = 1
- B. 21 ÷ 3 = 7
- C. 42 ÷ 2 = 21
- D. 84 ÷ 2 = 42
Write the letters in the right order
Answer:

13.216 cards are laid out in a rectangle, with the same number in each row and more than 1 row and more than 1 card a row. 216 = 2^□ × 3^□, so there are ways (a different number of rows counts as different).
Fill in each blank
Working:
Answer:
14.A rectangle has an area of 299 cm², and its length and width are both prime numbers of cm. What is its perimeter?
Working:
Answer:
15.A cube-shaped box holds 13824 cm³. How long is each edge?
Working:
Answer:
16.A square flower bed has an area of 1600 m². How long is each side?
Working:
Answer:
17.216 = 2^□ × 3^□. To make 216 × k a square, the smallest positive whole number k = .
Fill in each blank
Working:
Answer:
18.A lock's code is the sum of the different prime factors of 2160. What is the code?
Working:
Answer:

Show the answers and working

Prime factorisation worksheet: Answers and working

1. 4, 1, 1
240 divides by 2 4 times, then by 3 1 times, then by 5 1 times. So 240 = 2⁴ × 3¹ × 5¹.
2. 13
78 = 2¹ × 3¹ × 13: its prime factors are 2, 3 and 13, the largest 13.
3. 2, 1, 2
300 divides by 2 2 times, then by 3 1 times, then by 5 2 times. So 300 = 2² × 3¹ × 5².
4. 4, 1, 1
Divide again and again: by 2 4 times, by 3 1 times, by 7 1 times.
5. B → D → A → C
Each step divides the one before's result, ending on 1: 60 = 2² × 3 × 5.
6. 23
Work out the known part first: 2¹ × 3¹ = 6. 138 ÷ 6 = 23, a prime, so □ is 23.
7. 4, 3, 1
No need to multiply out 2160 first: break 72 and 30 into primes separately, then add up how many of each prime there are.
8. 2, 3, 1
No need to multiply out 540 first: break 60 and 9 into primes separately, then add up how many of each prime there are.
9. D → C → B → A
Each step divides the one before's result, ending on 1: 84 = 2² × 3 × 7.

10. True
2⁴ × 3³ = 432.
11. C. 2⁵ × 3² × 5
1440 = 2⁵ × 3² × 5: every factor is prime, and they multiply back to 1440.
12. True
2³ × 3³ = 216.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 3, 3, 14
The rows can be any factor of 216 except 1 and 216. 216 has (3 + 1) × (3 + 1) factors; take away those two: 14 ways.
14. 72 cm
299 = 13 × 23, both prime, so it is 23 cm by 13 cm; perimeter = 2 × (13 + 23) = 72 cm.
15. 24 cm
In 13824's prime factors every index is a multiple of 3; divide each by 3 for the edge: 24 cm.
16. 40 m
Break 1600 into primes: every index is even. Halve each index and the side is 40 m (40 × 40 = 1600).
17. 3, 3, 6
216 = 2³ × 3³. A square has every index even, so each prime with an odd index needs one more: k = 6.
18. 10
2160 = 2⁴ × 3³ × 5: its different prime factors are 2, 3 and 5, adding up to 10. Each prime counts once.

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