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Make it solvable (Posted on 2005-05-05) Difficulty: 4 of 5
a, b, and x are positive integers such that

sqrt(a) + sqrt(b) = sqrt(x)

How many possible values of x less than or equal to 1000 are there?

See The Solution Submitted by Jer    
Rating: 3.6667 (3 votes)

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re(4): First thoughts... ...(spoiler?) | Comment 5 of 16 |
(In reply to re(3): First thoughts... ...(spoiler?) by Charlie)

To see why the numbers need not be perfect squares:

sqrt(a) + sqrt(b) = sqrt(x)


a + b + 2*sqrt(a*b) = x

Only a*b need be a perfect square, so any prime factors that they have in common will form a perfect square and only the prime factors that they do not share need to be present in the original a and b in even powers.

  Posted by Charlie on 2005-05-05 19:51:17
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