Prime and composite numbers worksheet

Part A: Working out and filling in

1. 1. Least prime factor of 25 = 5

   Smallest prime factor of 25: try 2, 3, 5, 7 in turn: 25 is odd, so not by 2; its digits add up to 2 + 5 = 7, not a multiple of 3; it ends in 5, 25 ÷ 5 = 5, so the smallest prime factor is 5

2. 2. Least prime factor of 76 = 2

   Smallest prime factor of 76: try 2, 3, 5, 7 in turn: 76 is even, 76 ÷ 2 = 38, so the smallest prime factor is 2

3. 3. Least prime factor of 57 = 3

   Smallest prime factor of 57: try 2, 3, 5, 7 in turn: 57 is odd, so not by 2; its digits add up to 5 + 7 = 12, a multiple of 3, 57 ÷ 3 = 19, so the smallest prime factor is 3

4. 4. Least prime factor of 35 = 5

   Smallest prime factor of 35: try 2, 3, 5, 7 in turn: 35 is odd, so not by 2; its digits add up to 3 + 5 = 8, not a multiple of 3; it ends in 5, 35 ÷ 5 = 7, so the smallest prime factor is 5

5. 5. Least prime factor of 9 = 3

   Smallest prime factor of 9: try 2, 3, 5, 7 in turn: 9 is odd, so not by 2; its digit sum is 9, a multiple of 3, 9 ÷ 3 = 3, so the smallest prime factor is 3

6. 6. Least prime factor of 91 = 7

   Smallest prime factor of 91: try 2, 3, 5, 7 in turn: 91 is odd, so not by 2; its digits add up to 9 + 1 = 10, not a multiple of 3; it doesn't end in 0 or 5, so not by 5; 91 ÷ 7 = 13, so the smallest prime factor is 7

7. 7. Least prime factor of 81 = 3

   Smallest prime factor of 81: try 2, 3, 5, 7 in turn: 81 is odd, so not by 2; its digits add up to 8 + 1 = 9, a multiple of 3, 81 ÷ 3 = 27, so the smallest prime factor is 3

8. 8. Least prime factor of 93 = 3

   Smallest prime factor of 93: try 2, 3, 5, 7 in turn: 93 is odd, so not by 2; its digits add up to 9 + 3 = 12, a multiple of 3, 93 ÷ 3 = 31, so the smallest prime factor is 3

9. 9. Next prime after 93 = 97

   The next prime after 93: 94: 94 is even, 94 ÷ 2 = 47; 95: it ends in 5, 95 ÷ 5 = 19; 96: 96 is even, 96 ÷ 2 = 48; 97 divides by none of 2, 3, 5, 7, so it's prime — the answer is 97

10. 10. Next prime after 39 = 41

    The next prime after 39: 40: 40 is even, 40 ÷ 2 = 20; 41 divides by none of 2, 3, 5, 7, so it's prime — the answer is 41

11. 11. Least prime factor of 85 = 5

    Smallest prime factor of 85: try 2, 3, 5, 7 in turn: 85 is odd, so not by 2; its digits add up to 8 + 5 = 13, not a multiple of 3; it ends in 5, 85 ÷ 5 = 17, so the smallest prime factor is 5

12. 12. Next prime after 9 = 11

    The next prime after 9: 10: 10 is even, 10 ÷ 2 = 5; 11 divides by none of 2, 3, 5, 7, so it's prime — the answer is 11

Part B: Word problems

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

1. 13. 40 stamps are put into equal groups with none left over. There must be more than 1 group, and more than 1 in each group. What is the smallest number of groups possible? (Answer: 2 groups)

   The smallest prime factor of 40 is 2: 40 = 2 × 20

2. 14. Write 96 as a product of primes. What is the largest of its prime factors? (Answer: 3)

3. 15. Ka-lok's house number is the smallest prime number greater than 90. What is the house number? (Answer: 97)

   Checking the numbers after 90 one by one, the first prime is 97

4. 16. Tsz-ching is thinking of the largest prime number less than 47. What is the number? (Answer: 43)

   Checking the numbers below 47 one by one, the first prime is 43

5. 17. Ming is thinking of the largest prime number less than 29. What is the number? (Answer: 23)

   Checking the numbers below 29 one by one, the first prime is 23

6. 18. May's house number is the smallest prime number greater than 90. What is the house number? (Answer: 97)

   Checking the numbers after 90 one by one, the first prime is 97