document.write( "Question 584740: How many 2 digit numbers exist such that the product of the digits is greater than the actual number? \n" ); document.write( "
Algebra.Com's Answer #372880 by Edwin McCravy(20060)\"\" \"About 
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document.write( "There are no such two-digit numbers\r\n" );
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document.write( "Suppose that:\r\n" );
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document.write( "      tu > 10t + u\r\n" );
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document.write( "  tu - u > 10t\r\n" );
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document.write( "u(t - 1) > 10t\r\n" );
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document.write( "The tens digit is never 0.\r\n" );
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document.write( "Case 1:\r\n" );
document.write( "When the tens digit is 1, the number is \r\n" );
document.write( "10×1 + u or 10+u\r\n" );
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document.write( "and the product of digits is\r\n" );
document.write( "1×u or u, \r\n" );
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document.write( "and u is always less than 10 + u\r\n" );
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document.write( "So there are no two-digit numbers\r\n" );
document.write( "beginning with 1 whose product of\r\n" );
document.write( "digits is greater than the number.\r\n" );
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document.write( "Case 2:\r\n" );
document.write( "When the tens digit is 2 or more then\r\n" );
document.write( "t-1 is positive and we can divide the\r\n" );
document.write( "inequality\r\n" );
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document.write( "u(t - 1) > 10t\r\n" );
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document.write( "by t-1 and this inequality will hold:\r\n" );
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document.write( "            10t\r\n" );
document.write( "       u > ————— \r\n" );
document.write( "            t-1\r\n" );
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document.write( "Subtract then add 10 to the numerator:\r\n" );
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document.write( "            10t-10+10\r\n" );
document.write( "       u > ——————————— \r\n" );
document.write( "               t-1\r\n" );
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document.write( "Factor 10 out of the first two terms on top:\r\n" );
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document.write( "            10(t-1)+10\r\n" );
document.write( "       u > ——————————— \r\n" );
document.write( "               t-1\r\n" );
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document.write( "Break into two fractions:\r\n" );
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document.write( "            10(t-1)    10\r\n" );
document.write( "        u > ——————— + ————— \r\n" );
document.write( "              t-1      t-1\r\n" );
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document.write( "Cancel the t-1's in the first fraction:\r\n" );
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document.write( "                  10\r\n" );
document.write( "        u > 10 + ————— \r\n" );
document.write( "                  t-1\r\n" );
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document.write( "That is clearly false since the right side is greater than 10 and\r\n" );
document.write( "no units digit can be greater than 9.\r\n" );
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document.write( "This proves that no 2 digit numbers exist such that the product of \r\n" );
document.write( "the digits is greater than the actual number. \r\n" );
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document.write( "Edwin
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