document.write( "Question 849902: Digits a, b, and c can be chosen to make the following
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Algebra.Com's Answer #511997 by Edwin McCravy(20055) $\"\"$ $\"About$

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Digits a, b, and c can be chosen to make the following
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document.write( "        a b c\r\n" );
document.write( "×         2 4\r\n" );
document.write( "1  c  b  a  2\r\n" );
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document.write( "It's obvious that c must be either 8 or 3, since 8x4=32 and 4x3=12,\r\n" );
document.write( "the only ways to get a last digit of 2.\r\n" );
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document.write( "We try c=8\r\n" );
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document.write( "Simplify\r\n" );
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document.write( "Divide through by 10\r\n" );
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document.write( "Even if a and b were both the greatest possible digits, which\r\n" );
document.write( "is 9, the left side would be only 23*9+14*9 = 333, far short\r\n" );
document.write( "of 1781.  Thus we have ruled out c=8 as possible.  So c=3\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "Divide through by 10\r\n" );
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document.write( "We solve this Diophantine equation:\r\n" );
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document.write( "Since 14 is the coefficient with the smallest\r\n" );
document.write( "absolute value, 14, we write the 239 and the\r\n" );
document.write( "1293 in terms of their closest multiple of 14.\r\n" );
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document.write( "239/14 = 17.071...\r\n" );
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document.write( "So the closest multiple of 14 to 239 is 14*17=238\r\n" );
document.write( "and so 239 = 238+1\r\n" );
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document.write( "1293/14 = 92.357...\r\n" );
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document.write( "So the closest multiple of 14 to 1293 is 14*92=1288\r\n" );
document.write( "and so 1293 = 1288+5\r\n" );
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document.write( "So\r\n" );
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document.write( "becomes\r\n" );
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document.write( "Divide every term through by 14\r\n" );
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document.write( "Get all the fractions on the left side,\r\n" );
document.write( "and other terms on the right side:\r\n" );
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document.write( "The right side equals to an integer, so the left side\r\n" );
document.write( "must also equal to that same integer.  Let that \r\n" );
document.write( "integer be N, so:\r\n" );
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document.write( " and \r\n" );
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document.write( "Substitute in \r\n" );
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document.write( "Since b is a digit, the only integer\r\n" );
document.write( "N can be is 0. So N=0 and\r\n" );
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document.write( "Substituting N=0 and b=7 in\r\n" );
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document.write( "So a=5, b=7 and c=3\r\n" );
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document.write( "and the three digit number abc is 573\r\n" );
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document.write( "        5 7 3\r\n" );
document.write( "×         2 4\r\n" );
document.write( "1  3  7  5  2\r\n" );
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document.write( "Edwin

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