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problem 220 mathcounts

label Mathematics
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There exist two non congruent right triangles for which the length of the shorter leg in each triangle is 9 units and all sides have integer lengths. What is the sum of the lengths of the longer legs of these two triangles?

Sep 14th, 2014

that means, there are two sets of triangular numbers satisfying 

(i) a^2+ 81 = c^2  where a and c are integers 

(ii)x^2+81 = z^2  where x and z are integers.

also, in both cases, b^2 = 81 and y^2 = 81 are such that b is the least in a,b, and c which y is the least in x,y , and z.

this is not satisfied by any set of triangular numbers.

in other words, such 'b' and 'y' are not possible by number theory results. 

Sep 14th, 2014

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Sep 14th, 2014
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