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If you're familiar with the term 'difference of squares', as written in the name of this thread, you already know most of what you need to solve these!
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If X and Y are two positive integers such that the product of their sum and difference is 40, what is the value of X-Y?
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The product of their sum and difference is (x + y)(x - y), which equals the difference of squares, or \(x^2 - y^2\). In other words, \(x^2 - y^2=40\).
From here, guess and check to find two positive integers whose squares are 40 apart from each other. I actually found two different pairs.
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If X is the difference between the squares of two non-consecutive integers then is X prime?
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If x is the difference of squares, then you can rewrite x as \(a^2 - b^2\), which can be written as \((a-b)(a+b)\). In that case, is x prime?
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