Question

show that the square of positive integer is either of the form 6 Q + 1 or 6q + 3

Answer 1

let the remainders be 0,1,2,3,4,5
r=0
=6
r=1
(6q+1)2
36q2+1-12q
6q(6-2)+1
6q+1

r=2
(6q+2)2
36q2+4-24q
6q(6-4) +4
6q+4

r=3
(6q+3)2
36q2+9-36q
6q(6-6)+9
6q+9

r=4
(6q+4)2
36q2+16+48q
6q(6-8)+16
6q+16

r=5
(6q+5)2
36q2+25-60q
6q(6-10)+25
6q+25