Question

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

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

Assuming k is an odd integer, aye?

Answer 7

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.