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Solve the following question and mark the best possible option.

Find 3x^{2}y^{2} if x and y are integers such that y^{2} + 3x^{2}y^{2} = 30x^{2} + 517.

#### Options

612

418

588

None of these

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#### Solution

If we move the x^{2} term to the left side, it is factorable:

(3x^{2} + 1) (y^{2 }– 10) = 517 – 10 = 507 is equal to 3 x 13^{2}.

Since x and y are integers, 3x^{2} + 1 cannot equal a multiple of 3.

169 doesn't work either, so 3x^{2 }+ 1 = 13, and x = ± 2.

This leaves y^{2} – 10 = 39, so y = ± 7.

Thus, 3x^{2}y^{2} = 3 x 4 x 49 = **588**.

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