Question

# If a, b, c are the three natural numbers such that a + b + c is divisible by 6, then a^{3} + b^{3} + c^{3} must be divisible by -

- 6
- 12
- 24
- None of these

## The correct answer is: 6

### We have (a^{3} – a) + (b^{3} – b) + (c^{3} – c)

= (a – 1) (a) (a + 1) + (b – 1) (b) (b + 1) + (c – 1) (c) (c + 1)

Since (x – 1) x (x + 1) is divisible both 2 & 3, it is divisible by 6. Thus,

6 | {(a^{3} – a) + (b^{3} – b) + (c^{3} – c)}

6 | (a^{3} + b^{3} + c^{3}) as 6|(a + b + c).

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