Since the number you start with is odd you can write it 2n+1, that is, it is one more than an even number. Squaring 2n+1 gives
Look at it like this:
(2n+1)^2=
4n^2+4n+1=
4(n^2+n)+1=
4n(n+1)+1
Thus the square of 2n+1 is one more than 4n(n+1). Since n and n+1 are consecutive integers one of them must be even, and hence 2 divides n(n+1). Thus 8 divides 4n(n+1) and hence 4n(n+1)+1 is one more than a multiple of 8. That is dividing 4n(n+1)+1 by 8 leaves a remainder of 1.
Does this help?
+98 
