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Divisibility to square (Posted on 2011-03-05) Difficulty: 3 of 5
Each of x and y is a positive integer such that x2 + y2 + x is divisible by 2xy.

Prove that x is the square of an integer.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Suggested solution | Comment 1 of 4
If (x^2+y^2+x)=2kxy, k>0, then by adding 2xy-x to both sides and factoring, one gets (x+y)^2=((2k+2)y+1)x

If a prime p divides x, then it must divide the left side x^2+2xy+y^2, and so p divides y^2 and thus y.

But then this implies ((2k+2)y+1) is not divisible by p, and thus since (x+y)^2 must have an even power of p in its prime factorization, so must x.

Since this applies for every prime p dividing x, this implies x must be the square of some integer.

  Posted by Gamer on 2011-03-08 03:53:11
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