Take any odd number and square it. It will invariably be a multiple of 8 plus 1. So (odd)^2=8n+1 where n is an integer. Show why this is always so. Also show what the pattern for n is.

induction is the shortest proof:

for n=1 8*1+1 ok

given the formula holds for n

for n+2 (next odd) 8*( n+2)+1= (8*n+1)+16

qed

ady