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Statement - 1: For each natural number n, (n+1)⁷ - n⁷ is divisible by 7. Statement - 2: For each natural number n, n⁷ - n is divisible by 7.

1. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
2. Statement-1 is true, Statement-2 is true; Statement-2 is NOT a correct explanation for Statement-1
3. Statement-1 is true, Statement-2 is false
4. Statement-1 is false, Statement-2 is true

Correct answer: Statement-1 is false, Statement-2 is true

Solution

(n+1)^7 - n^7 = 7n^6 + 21n^5 + 35n^4 + 35n^3 + 21n^2 + 7n + 1; the trailing +1 is not divisible by 7, so Statement-1 is false. Statement-2 (n^7 - n divisible by 7) is true by Fermat's little theorem. Hence S1 false, S2 true.

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