document.write( "Question 1067650: √a^2+b^2=613 where a, b are positive integers.
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Algebra.Com's Answer #682792 by Edwin McCravy(20064)\"\" \"About 
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\"sqrt%28a%5E2%2Bb%5E2%29\"\"%22%22=%22%22\"\"613\"
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document.write( "a² + b² = 613\r\n" );
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document.write( "We see if (a,b,613) is a Pythagorean triple\r\n" );
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document.write( "let b = 613-k\r\n" );
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document.write( "a² + (613-k)² = 613²\r\n" );
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document.write( "a² + 613² -1226k + k² = 613²\r\n" );
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document.write( "a² - 1226k + k² = 0\r\n" );
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document.write( "We notice that \"sqrt%281226%29=%2235.01142...%22\"\r\n" );
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document.write( "So the largest square not exceeding 1226 is 35² = 1225\r\n" );
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document.write( "So we write 1226 = 35² + 1\r\n" );
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document.write( "a² - (35²+1)k + k² = 0\r\n" );
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document.write( "a² = (35²+1)k - k² \r\n" );
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document.write( "a² = 35²k + k - k²\r\n" );
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document.write( "We see that if k=1 the right side becomes 352\r\n" );
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document.write( "Therefore k=1, and b = 613-k = 613-1 = 612, and a=35\r\n" );
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document.write( "Therefore (a,b,613) = (35,612,613) is a Pythagorean triple.\r\n" );
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document.write( "Thus a+b = 35+612 = 647\r\n" );
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document.write( "Edwin
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