document.write( "Question 543124: Supose that a right triangle has sides a, b, and c which happen to be Natural numbers with c the largest. if you are told nothing about whether a, b, or c are odd or even then can it be determined whether the number (a+b+c) is odd or even justify your answer. \n" ); document.write( "

Algebra.Com's Answer #354758 by richard1234(7193) $\"\"$ $\"About$

You can put this solution on YOUR website!
By looking at certain right triangles such as 3-4-5, 5-12-13, 8-15-17, etc., we can conjecture that a+b+c is always even. To prove this, suppose that the opposite was true, that a+b+c could be odd. If this is so, then either all three are odd (contradiction, because odd + odd is not equal to odd) or exactly one of the three is odd (also contradiction because even + odd is never even, even + even is never odd). Hence a+b+c is even.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Tip: Instead of using odd + odd or even + even, use something like \"(2k+1) + (2m+1)\" or go even simpler and use modular arithmetic (e.g. 1+1 is never congruent to 1 mod 2). \n" ); document.write( "

\n" );