Question

Show that the square of any positive integer can't be of form 6m+2 or 6m+5 for any integer m.

Answer 1

let a be any positive integer then b=3
where 0>r>3
Case I
when r=0
a=bq+r
3q+0
=(3q.)sq.=(9q sq.)
Case II
when r= 1
(3q+1) sq.
9q sq.+1+6q
3(3q+2q+1)
3m. where m=3q+2q+1
case III
where r =2
(3q+2) sq
9q sq+4+12q
3(3q sq +4q) +4
3m+4. where m=3q sq+4q)


hence it proves that the square of any +ve integer cant be of the form 6m+2. 6m+5