Question
If a, b, c are the three natural numbers such that a + b + c is divisible by 6, then a3 + b3 + c3 must be divisible by -
- 6
- 12
- 24
- None of these
The correct answer is: 6
We have (a3 – a) + (b3 – b) + (c3 – 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 | {(a3 – a) + (b3 – b) + (c3 – c)}
6 | (a3 + b3 + c3) as 6|(a + b + c).
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