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Find all positive integers n such that 12n-119 and 75n-539 are both perfect squares. Let N be the sum of all possible values of n. Find N.

Let 75n - 539 = l^2 and 12n - 119 = k^2 . where n is a natural number. Multiply 75n - 539 = l^2 by 4 to give 300n - 2156 =4l^2 and 12n - 119 = k^2 by 25 to give 300n - 2975 = 25k^2. Subtract the two new expressions to give 4l^2 - 25k^2 = 819 which can be factorised (using the difference of two squares) to give (2l - 5k)(2l + 5k) = 819. The prime factorisation of 819 is 3^2 * 7 * 13 There are five cases to consider. Dealing with the cases (noting that 2l - 5k < 2l +5k ) yields that n can only be 20 or 12. Hence N = 20 + 12 = 32.
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