A number theory problem by Jose Sacramento

Number Theory Level 2

What is the minimum value of the sum of two positive integers whose product is 36?

20 15 37 12

Jose Sacramento by Jose Sacramento , 66, Portugal

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1 solution

Alekhya China
Sep 16, 2016

for x+y=k. x*y is minimum for x=y=k/2 .

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Don't you have it backwards? We want to minimize X + y, not x y. I think you mean for x y = k, x + y is minimum for x = y = sqrt(k). Think of it this way: Let x*y = k Then x + y = x + k/x. Differentiating with respect to x and setting the result equal to 0 gives 1 - k/(x^2) = 0, or x^2 = k, so x + sqrt(k), and y = k/x = k/sqrt(k) = sqrt(k). Ed Gray

Edwin Gray - 2 years, 5 months ago

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