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Pythagorean areas (Posted on 2017-11-17) Difficulty: 3 of 5
List all Pythagorean triangles whose area utilizes
9 different digits.

See The Solution Submitted by Ady TZIDON    
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observations | Comment 4 of 7 |
One leg of a PT will always be divisible by 3, and one (possibly the same leg) will always be divisible by 4 so area will be divisible by 6.

Since the sum of the digits = 45, only 0,3,6,9 can be the omitted digit.

Using the parametric solution for primitive solutions, x=2ab and y=a^2-b^2 with (a,b) relatively prime and of opposite parity.   Then area = ab(a+b)(a-b), the product of four relatively prime numbers.

For non-primitive solutions the legs are scaled by a common factor so area will be divisible by a square. 

None of this points to a solution, not to me anyway, but at least it limits the search space.

  Posted by xdog on 2017-11-18 07:37:04
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