Prove by direct method that for any integer ′n ′ , n3 − n is alw – InterviewSolution

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1.

Prove by direct method that for any integer ′n ′ , n3 − n is always even.

Answer»

Let n be even, n = 2m

∴ n³ − n = n(n² − 1) = 2m (4m² − 1), which is even.

If n is odd, n = 2m + 1.

Then, n³ − n = (2m + 1)³ − (2m + 1)

= (2m + 1){4m² + 4m + 1 − 1}

= 4(m² + m) (2m + 1), which is also even.

∴ For any integer n, n³ − n is always even.

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