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$a$ | $45a$ | Square? |
---|---|---|
$1$ | $45$ | No |
$2$ | $90$ | No |
$3$ | $135$ | No |
$4$ | $180$ | No |
$5$ | $225$ | Yes: $15^2 = 225$ |
This shows that the smallest possible number would be $5$.
Alternatively, for $45a$ to be a square number, each of its prime factors must be raised to an even power. Since $45 = 3^2\times 5$ as a product of prime factors, the only prime factor not raised to an even power is $5$. Therefore the smallest value of $a$ that makes this a square must be $5$.