Question

using euclid lemma to show that square of an odd positive integer is of the form 8n+1,where n is any integer

Answer 1

let a and b be two positve odd integer with a<b

where b=8

possible remainders are 0,1,2--------7

a=8q+0----1

a=8q+1-----2

-------a=8q+7----6

now taking square of all above values

a2=8q2+1

=[64q2]+1+16q

=8[8q2+2q]+1

=8n+1         {by replaceing 8q2+2q}by n