Question

prove that square of any integer is either of the form 3k or 3k+1
prove this using division algorithm

Answer 1

1^2?

Answer 2

use the fact that the square of a number its either in the form 2n or 2n+1 (or 2n*2n or (2n+1)^2), Or use n^2 which is n*n and show that 3k|n or 3k+1|n

Answer 3

state division algo: Let a,b be integers with b>0. Then there exists unique integers q and r such that a = bq + r and 0<=r<b

Answer 4

a^2=(3q)^2=9q^2=3(3q^2), or
a^2=(3q+1)^2=9q^2+6q+1 = 3(3q^2 + 2q) + 1, or
a^2=(3q+2)^2=9q^2+12q+4 = 3(3q^2 + 4q + 1) + 1