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The Power(ful) Equations (Posted on 2007-12-22) Difficulty: 2 of 5
If x and y are positive integers, L: LCM of x and y, G: GCD of x and y, then solve the following equations for x and y values:
1) xy=LG, L>x.
2) x2+y2 = L2+G2, L>x>G.

See The Solution Submitted by Praneeth    
Rating: 4.0000 (2 votes)

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Solution Solution, Part 2 | Comment 1 of 4
Let a = x/G, b = y/G

Then L = Gab.

Substituting into equation 2 yields

(Ga)² + (Gb)² = (Gab)² + G²

solving yields a = 1, b = 1 (if a,b are positive)

But then L = G, which violates L > x > G,

so there is no solution




        

  Posted by Steve Herman on 2007-12-22 14:51:16
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