Use induction to show that 8 divides n2-1 for every odd integer n. 2k+1 Use cong
ID: 3080300 • Letter: U
Question
Use induction to show that 8 divides n2-1 for every odd integer n. 2k+1 Use congruence to show that 8 divides n2-1 for every odd integer n.Explanation / Answer
Prove By Induction: 8 divides n^2-1 whenever n is a non-negative integer. First not that f(n) = n^2-1 = (n+1)(n-1) = (n-1)(n+1) For the case n = 1: f(1) = (1 - 1)(1+1) = 0 = 8*B, for B = 0 so => 8 divides n^2-1 , for n = 1. Now assume that 8 divides f(n) for n=1..k, now need to show that f(k+1) is also divisible by 8. f(k+1) = ([k+1]-1)([k+1]+1) = k(k+2) = (k-1+1)(K+1+1) = f(k)+(k+1)+(k-1)+1 =f(k)+2k Since 2k is always an even number, so can be written as 2L for some integer L => f(k+1) = 8B + 2L = 8M Since B and L are both integers, their sum M is an integer => f(k+1) is divisible by 8. Therefor by the induction hypothesis f(n) is divisible by 8 for all non-negative n.
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