Types of Numbers
Mark Scheme & Model Answers
✓ ANSWERS
Section A — Odd and Even Numbers (Q1–5) [5 marks]
Q1. 42, 100, 4,836, 50,020 [1 mark]
Q2(a). Odd + Odd = Even
Q2(b). Even × Odd = Even
Q2(c). 7 + 13 + 6 = Even (= 26)
Q2(d). 3 × 5 × 4 = Even (= 60) [1 mark]
Q3. Largest even number < 1,000 998
Q3. Smallest odd number > 9,990 9,991 [1 mark]
Q4. 111, 333, 555, 777, 999 [1 mark]
Q5. Odd. Even + Odd = Odd. Since 47 is odd and one number is even, the other must be odd. [1 mark]
Section B — Prime Numbers (Q6–10) [8 marks]
Q6. 11, 13, 17, 19, 23, 29 [1 mark]
Q7(a). 29 — Factors: 1, 29 → Prime
Q7(b). 33 — Factors: 1, 3, 11, 33 → Not prime
Q7(c). 51 — Factors: 1, 3, 17, 51 → Not prime
Q7(d). 61 — Factors: 1, 61 → Prime [2 marks total]
Q8. 1 has only one factor (itself). A prime number must have exactly two distinct factors: 1 and itself. Since 1 only has one factor, it does not qualify. [1 mark]
Q9. 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3 [2 marks]
Q10. 3 + 17 = 20 | 7 + 13 = 20 [2 marks]
Section C — Square Numbers (Q11–15) [5 marks]
Q11(a). 6² = 36
Q11(b). 9² = 81
Q11(c). 12² = 144
Q11(d). 15² = 225 [1 mark]
Q12. 16, 25, 36, 49, 81, 100 [1 mark]
Q13(a). √64 = 8
Q13(b). √121 = 11
Q13(c). √144 = 12 [1 mark]
Q14. 14 m (√196 = 14) [1 mark]
Q15. False. Example: 9 = 3² is odd. Odd × odd = odd, so odd numbers squared give odd square numbers. [1 mark]
Section D — Cube Numbers (Q16–20) [5 marks]
Q16(a). 2³ = 8
Q16(b). 4³ = 64
Q16(c). 5³ = 125
Q16(d). 10³ = 1,000 [1 mark]
Q17. 1, 8, 27, 64, 125, 216 [1 mark]
Q18(a). ∛27 = 3
Q18(b). ∛125 = 5
Q18(c). ∛1,000 = 10 [1 mark]
Q19. 8 cm (8 × 8 × 8 = 512) [1 mark]
Q20. Yes — 10 × 10 × 10 = 1,000, so 1,000 is a cube number. ∛1,000 = 10 [1 mark]
Section E — Mixed Challenge (Q21–25) [11 marks]
Q21. Any of: 4, 16, 36, 64, 100…
Q21. Any of: 1, 9, 25, 49… [2 marks]
Q22. 2. All other even numbers are divisible by 2 (as well as by 1 and themselves), so they have at least three factors and cannot be prime. [2 marks]
Q23. 1 (1² = 1 and 1³ = 1). These numbers are called perfect sixth powers (n&sup6;). The next one, 64 = 2&sup6;, lies outside the range 1–30. [2 marks]
Q24. Valid pair: (4, 3) — 4 − 1 = 3, which is prime.
Checking all: 1−1=0 no; 4−1=3 yes; 9−1=8 no; 16−1=15 no; 25−1=24 no; 36−1=35 no; 49−1=48 no.
Only valid pair: (square 4, prime 3) [2 marks]
Q25. First four cube numbers — also square? Also prime? Even or odd?
| Cube number | Also a square? | Also prime? | Even or odd? |
|---|
| 1³ = 1 | Yes (1² = 1) | No (only 1 factor) | Odd |
| 2³ = 8 | No | No | Even |
| 3³ = 27 | No | No | Odd |
| 4³ = 64 | Yes (8² = 64) | No | Even |
[3 marks]