Show that n^2- 1 is divisible by 8 for all positive odd numbers n

Question

terenzreignz · Answer

Induction? That involves two steps... first one being... show that it holds for n = 1.

terenzreignz · Answer

(come on, this is the simplest step ;)  )

anonymous · Answer

yeah after stating the induction hypothesis, and all that, i get (k^2 +2k +1) -1 is div by 8 ...not sure how to prove that

terenzreignz · Answer

Hang on...
Assume k^2 - 1 is divisible by 8
Then let's have a look at
$$\Large (k+1)^2- 1$$

terenzreignz · Answer

Or rather...
$$\Large (k+2)^2-1$$

terenzreignz · Answer

Assuming k is an odd integer, aye?

anonymous · Answer

Since n is odd n=4m+1 or n=4m+3.

In the first case n2−1=(n−1)(n+1)=4m⋅(4m+2)=8m(2m+1), while in the second case n2−1=(n−1)(n+1)=(4m+2)⋅(4m+4)=8(2m+1)(m+1).
So n2−1 is divisible by 8 if n is odd.