Question

show that the square of any positive integer is of the form 3M or 3M + 1 for some integer m

Answer 1

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Answer 2

Let a be any positive integer
applying Euclids Division Lemma for a and 3
we have
a=3q+r
where r=0, 1 and 2
r=0 which implies a=3q +0=3q
a2= (3q) 2= 9q2= 3×3q2
considering 3q2 as m, it will be
a2 = 3m

r=1 which implies a = 3q + 1
a2 = (3q + 1) 2 = 9q2 +6q + 1
= 3 ( 3q2 + 2q ) +1
considering 3q2 + 2q as m, we can say that
a2 = 3m + 1