Question

Product of three different positive integers is 144 what is the maximum possible sum of these integers

Answer 1

Algebraically speaking, this is tricky. But not too much. Normally, to reduce a system of equations with n unknowns, you need at least n # equations. We have two:
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x * y * z = 144
x^2 + y^2 + z^2 = 149
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We also have the condition that x, y, and z are all positive integers. This helps.
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The problem wants us to find the sum of x + y + z
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This can be solved with logic and reasoning, and some trial and error. I used an MS Excel spreadsheet to speed up the calculations (so I could focus on the logic), but you could do it just as easily with pencil and paper.
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The easiest place to start is with the products of x * y * z = 144
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144 = 12^2
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If we only had two integers to multiply, starting with 12 * 12 is a great start. But we have three. So, factoring 12 and something like 3 * 4 * 12 makes a good guess.
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This gets us x * y * z = 144 okay, but not the sum of the squares:
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3^2 + 4^2 + 12^2 = 169, which is greater than the 149 we are looking for. But that gives us another clue:
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Keeping under the sum of squares constraint with an integer as large as 12 is probably impossible. So, we are narrowing down the possible solutions.
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Next tactic, factor 144:
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1, 144

Answer 2

Answer:

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