Question

use euclids division lemma to show that the square of any postive integer is of the form 3p 3p+1

Answer 1

a=bq+r 0_<r <b
b=3 r=0,1,2
r=0
a=3q+0
square on both side
a2 = 9q2
a2=3 (3q2) p=3q2
a2=3p
r=1
a=3q+1
square on both side
a2=9q2+1+6q
a2=3 (3q2+2q)+1 p=3q2+2q
a2=3p+1
r=2
a=3q+2
square on both side
a2=9q2+4+12q
a2=9q2+3+1+12q
a2=3 (3q2+1+4q)+1 p=3q2+1+4q
a2=3p+1