12
Question Determine whether the statements below are ‘Always True’, ‘Sometimes True’, or ‘Never True’. Explain your reasoning.
(i) The product of a multiple of 6 and a multiple of 3 is a multiple of 9.
(ii) The sum of three consecutive even numbers will be divisible by 6.
(iii) If abcdef is a multiple of 6, then badcef will be a multiple of 6.
(iv) 8(7b – 3) – 4(11b + 1) is a multiple of 12.
(i) Always True, The multiple of 6 can be written as 6a, where a is an integer. The multiple of 3 can be written as 3b, where b is an integer. ∴ Product = (6a) × (3b) = 18(ab) is a multiple of 9. (ii) Always True, The sum of three consecutive even numbers will be divisible by 6. For example 2 + 4 + 6 = 12, 4 + 6 + 8 = 18, 6 + 8 + 10 = 24, 8 + 10 + 12 = 30,… These numbers are divisible by 6. (iii) Always True, because one’s place does not change. (iv) Sometimes true, Conclusion: 8(7 × 1 – 3) – 4(11 × 1 + 1) = -16, not divisible by 12. 8(7 × 10 – 3) – 4(4 × 10 + 1) = 536 – 164 = 372, divisible by 12.