Question

show that the square of an odd positivel integer is of the form 8n+1 for some integer n

Answer 1

Let a be any odd positive integer with b=4.
Therefore, 
a= 4q+1 ( By Euclid's Division Lemma)
When a=4q+1
a2=(4q+1)2
=16q2 +1 + 8q
=8(2q2 +q) +1
=8q+1 where (q=2q2+q)
Hence proved.