perplexus.info :: Just Math : Root Difference and Seventeen Puzzle

Root Difference and Seventeen Puzzle (Posted on 2015-08-29)

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.

See The Solution Submitted by K Sengupta

Rating: 4.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

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

Similarly.

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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