document.write( "Question 53846: Hi guys! Some of you may already know me as aaaaaaaa, and I'm asking a question:\r
\n" ); document.write( "\n" ); document.write( "Solve the equation:\r
\n" ); document.write( "\n" ); document.write( " $\"j%5E2+-+%287c%29%5E2+=+99\"$
\n" ); document.write( "j and c are in N (natural numbers).\r
\n" ); document.write( "\n" ); document.write( "I know that $\"j%5E2+-+%287c%29%5E2+=+%28j%2B7c%29%28j-7c%29\"$ , but the question is: Is there a way to solve this without trial-and-error and find out that g=50, c=7? \n" ); document.write( "

Algebra.Com's Answer #36390 by AnlytcPhil(1806) $\"\"$ $\"About$

You can put this solution on YOUR website!

Hi guys! Some of you may already know me as \r\n" );
document.write( "aaaaaaaa, and I'm asking a question:\r\n" );
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document.write( "Solve the equation:\r\n" );
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document.write( "j² - (7c)² = 99\r\n" );
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document.write( "j and c are in N (natural numbers).\r\n" );
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document.write( "I know that j² - (7c)² = (j+7c)(j-7c), but the \r\n" );
document.write( "question is: Is there a way to solve this without \r\n" );
document.write( "trial-and-error and find out that j=50, c=7?\r\n" );
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document.write( "j² - (7c)² = 99\r\n" );
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document.write( "(j + 7c)(j - 7c) = 99\r\n" );
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document.write( "For that factorization of 99, the two factors,\r\n" );
document.write( "(j + 7c) and (j - 7c) differ by 14c,\r\n" );
document.write( "a multiple of 14\r\n" );
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document.write( "The only factors of 99 are 1, 3, 9, 11, 33, 99   \r\n" );
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document.write( "So there are only three ways to write 99 as the product \r\n" );
document.write( "of a larger natural number times a smaller natural number,\r\n" );
document.write( "namely these three ways\r\n" );
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document.write( "99×1, 33×3 and 11×9.\r\n" );
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document.write( "Of the three ways only the first are factors that differ\r\n" );
document.write( "by a multiple of 14, namely 98, since 98 = 14×7, so \r\n" );
document.write( "14c = 14×7 so \r\n" );
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document.write( "c = 7\r\n" );
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document.write( "and since j² - (7c)² = 99,\r\n" );
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document.write( "j² - (7×7)² = 99\r\n" );
document.write( "j² - 49² = 99\r\n" );
document.write( "j² - 2401 = 99\r\n" );
document.write( "j² = 2500\r\n" );
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document.write( "j = 50\r\n" );
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document.write( "Edwin

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