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Root Difference and Seventeen Puzzle (Posted on 2015-08-29) Difficulty: 3 of 5
Each of X and Y is a positive integer which is not a perfect square that satisfy:
√X - √Y = √17
Is each of X and Y always divisible by 17?
Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution Algebraic solution (spoiler) | Comment 2 of 3 |
sqrt(x) = sqrt(17) + sqrt(y)
squaring gives
x = 17 + y + 2sqrt(17y)
the LHS is integer. so the RHS is also, so y must be a multiple of 17

sqrt(y) = -sqrt(17) + sqrt(x)
squaring gives
y = 17 + x - 2sqrt(17x)
the LHS is integer. so the RHS is also, so x must be a multiple of 17

Note that this is true for any x and y, even if x or y is a perfect square

  Posted by Steve Herman on 2015-08-30 07:12:23
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