document.write( "Question 1169510: Consider integers x and y. When y is devided by x the remainder is 29. When devided by x/2, remainder is 13. Determine x. \n" ); document.write( "
Algebra.Com's Answer #794480 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The other tutor did not prove anything at all.  He found only one case\r\n" );
document.write( "where x = 32.\r\n" );
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document.write( "Let the quotient be n when y is divided by x\r\n" );
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document.write( "   n\r\n" );
document.write( "x) y\r\n" );
document.write( "  nx\r\n" );
document.write( "   y-nx      <---REMAINDER\r\n" );
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document.write( "So \r\n" );
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document.write( "y-nx = 29\r\n" );
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document.write( "y-29 = nx\r\n" );
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document.write( "\"%28y-29%29%2Fn=x\"  \r\n" );
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document.write( "Let the quotient be m when y is divided by x/2\r\n" );
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document.write( "    m  \r\n" );
document.write( "x/2) y\r\n" );
document.write( "    mx/2\r\n" );
document.write( "     y-mx/2      <---REMAINDER\r\n" );
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document.write( "So\r\n" );
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document.write( "y-mx/2 = 13\r\n" );
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document.write( " 2y-mx = 26\r\n" );
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document.write( " 2y-26 = mx\r\n" );
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document.write( "\"%282y-26%29%2Fm=x\"\r\n" );
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document.write( "Set the 2 expressions for x equal:\r\n" );
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document.write( "\"%28y-29%29%2Fn=%282y-26%29%2Fm\"\r\n" );
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document.write( "my - 29m = 2ny - 26n\r\n" );
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document.write( "my - 2ny = 29m - 26n\r\n" );
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document.write( "y(m - 2n) = 29m - 26n\r\n" );
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document.write( "\"y+=+%2829m-26n%29%2F%28m-2n%29\"\r\n" );
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document.write( "Divide that out as is:\r\n" );
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document.write( "     29     \r\n" );
document.write( "m-2n)29m-26n\r\n" );
document.write( "     29m-58n\r\n" );
document.write( "         32n      <---REMAINDER \r\n" );
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document.write( "So \"y=29%2B%2832n%29%2F%28m-2n%29\"\r\n" );
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document.write( "Divide that out the other way:\r\n" );
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document.write( "         13       \r\n" );
document.write( "-2n + m)-26n + 29m\r\n" );
document.write( "        -26m + 13m\r\n" );
document.write( "               16m      <---REMAINDER\r\n" );
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document.write( "So  \"y+=+13%2B%2816m%29%2F%28-2n%2Bm%29\"\r\n" );
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document.write( "\"y+=+29%2B%2832n%29%2F%28m-2n%29+=+13%2B%2816m%29%2F%28-2n%2Bm%29\"\r\n" );
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document.write( "\"y%28m-2n%29=29%2B32n=13%2B16m\"\r\n" );
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document.write( "\"29%2B32n=13%2B16m\"\r\n" );
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document.write( "\"16=16m-32n\"\r\n" );
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document.write( "\"1=m-2n\"\r\n" );
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document.write( "So\r\n" );
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document.write( "\"y=29%2B%2832n%29%2F%28m-2n%29\"\r\n" );
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document.write( "\"y=29%2B%2832n%29%2F%281%29\"\r\n" );
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document.write( "\"y=29%2B32n\"\r\n" );
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document.write( "Since \"%28y-29%29%2Fn=x\"\r\n" );
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document.write( "\"%2829%2B32n-29%29%2Fn=x\"\r\n" );
document.write( "\"32n%2Fn=x\"\r\n" );
document.write( "\"32=x\"\r\n" );
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document.write( "So x=32      <---ANSWER\r\n" );
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document.write( "y can vary according to this rule: \r\n" );
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document.write( "\"y=29%2B32n\"\r\n" );
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document.write( "but x can only be 32.\r\n" );
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document.write( "There may be shorter ways, but this proves it.\r\n" );
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document.write( "Edwin
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