n2 – 1 is divisible by 8, if n is ______.

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KollegeApply · Accepted Answer

n2 – 1 is divisible by 8, if n is an odd integer. Explanation: Let x = n2 – 1 In the above equation, n can be either even or odd. Let us assume that n = even. So, when n = even i.e., n = 2k Where k is an integer We get, \implies x = (2k)2 – 1 \implies x = 4k2 – 1 At k = – 1, x = 4(–1)2 – 1 = 4 – 1 = 3, is not divisible by 8. At k = 0, x = 4(0)2 – 1 = 0 – 1 = – 1, is not divisible by 8 Let us assume that n = odd: So, when n = odd i.e., n = 2k + 1 Where k is an integer We get, \implies x = 2k + 1 \implies x = (2k + 1)2 – 1 \implies x = 4k2 + 4k + 1 – 1 \implies x = 4k2 + 4k \implies x = 4k(k + 1) At k = –1, x = 4(–1)(–1 + 1) = 0 which is divisible by 8. At k = 0, x = 4(0)(0 + 1) = 0 which is divisible by 8. At k = 1, x = 4(1)(1 + 1) = 8 which is divisible by 8. From the above two observation We can conclude that, if n is odd, n2 – 1 is divisible by 8.

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