Question

use euclid s division lemma to show that the sq root of any positive integer is of the form 3p,3p+1

Answer 1

let a be any positive integer and b=3
by euclid's division lemma,a=bq+r,
(r=0,1,2)
whn r=0,
a=3p+0=3p
a^2=3p^2=9p^2

when r=1
a=3p+1
a^2=(3p+1)^2=(3p)^2+2*3p*1+1^2
=9p^2+6p+1
3 (3p^2+2p)+1=3p+1

whn r=2
do by usin (a+b)^2

this implies tht the sq of ny +integer is of th form 3p,3p+1

hope it helps