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Fractions lead to Pythagoreans (Posted on 2016-06-28) Difficulty: 2 of 5
Write down two fractions whose product is 2.

Add 2 to each. Keep them improper.

Cross multiply to get two whole numbers.

These numbers are the legs of a Pythagorean triangle!

Prove this always works.

No Solution Yet Submitted by Jer    
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solution | Comment 1 of 4
The two fractions satisfying the given conditions must have the form p/q and 2q/p, where p and q are positive integers.

So adding 2 to each and simplifying, we have:
(2q+p)/q and 2(p+q)/p
Cross multiplying, we have:
p(2q+p) and 2q(p+q)
or, 2pq+p^2 and 2q(p+q)--(*)

Substituing p+q=x,
we see that: x > q as p+q > q  
and, (*) reduces to:
x^2 - q^2 and 2qx which is the well known form for the two legs of a pythagorean triangle.
  Posted by K Sengupta on 2016-06-28 11:36:40
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