Discussion
Square of any integer is of the form :
Answer: D
A square is of the form \(3k\) or \(3k - 2\).
Any integer is \(3m\), \(3m + 1\) or \(3m - 1\), and
\[ (3m)^{2} = 9m^{2},\quad (3m \pm 1)^{2} = 9m^{2} \pm 6m + 1 \]
So a square leaves remainder 0 or 1 on division by 3. Remainder 1 can be written \(3k - 2\) (with \(k = m' + 1\)), so squares are \(3k\) or \(3k - 2\).
The other options fail: \(3k - 1\) means remainder 2, impossible; \(4k - 1\) means remainder 3 mod 4, but squares are 0 or 1 mod 4; and \(5k\) or \(5k + 1\) misses squares like 4 and 9, which are 4 mod 5.
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