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jimmypoopins
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Homework Statement
Prove that the square of any integer a is either of the form 3k or of the form 3k+1 for some integer k.
Homework Equations
The Division Algorithm: 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
The Attempt at a Solution
I know from the division algorithm that any integer a can be written as 3q, 3q+1, or 3q+2, so
a^2=(3q)^2=9q^2=3(3q^2), or
a^2=(3q+1)^2=9q^2+6q+1, or
a^2=(3q+2)^2=9q^2+12q+4,
but i don't understand how the 3q+1 or 3q+2 case helps me.
Can anyone give me some hints or point me further in the right direction? thanks.
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