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 Big areas (Posted on 2013-09-17)
How many Pythagorean triangles are there whose area is expressed by 9 different digits?

Provide the values for the smallest and the biggest of them.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 quick observations | Comment 1 of 16
The legs (x,y) of a primitive pythagorean triangle are x=2ab and y=a^2-b^2 where (a,b) are relatively prime and of opposite parity.  All other PTs can be obtained from primitive values by multiplying x and y by any integer k.

The area is one-half the product of the legs or Area = k^2 * ab * (a^2 - b^2).  This is always even since one of (a,b) is even.  Also, looking at values of a and b mod 3 shows area is divisible by 3 as well.

The sum of 0,1, ... 9 is 45 which is divisible by 3 so the excluded digit in the sought for area can only be 0,3,6,9.

 Posted by xdog on 2013-09-17 17:56:04

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