If y is the smallest positive integer such that multiplying 3150 by y gives the square of an integer, what is the value of y?
14
14 is correct.
A positive integer is a perfect square exactly when every prime in its factorisation appears to an even power. To make \(N \times y\) a perfect square with the smallest y, we take the primes of N that appear to an odd power and multiply them together once each — this "pairs up" every odd exponent.
Factorising 3150 gives \(3150 = 2^1 \times 3^2 \times 5^2 \times 7^1\). The primes 3 and 5 already appear to even powers; 2 and 7 appear to odd powers. So the smallest y is \(2 \times 7 = 14\), and \(3150 \times 14 = 44100 = 210^2\) confirms the result.
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