document.write( "Question 265927: Find the smallest positive integer n such that the product of n and 999 does not contain the digit 9.\r
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Algebra.Com's Answer #195462 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "999*1=999\r\n" );
document.write( "999*2=1998\r\n" );
document.write( "999*3=2997\r\n" );
document.write( "999*4=3996\r\n" );
document.write( "999*5=4995\r\n" );
document.write( "999*6=5994\r\n" );
document.write( "999*7=6993\r\n" );
document.write( "999*8=7992\r\n" );
document.write( "999*9=8991\r\n" );
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document.write( "So n cannot be a 1 digit number\r\n" );
document.write( "since every product of a positive digit and 999\r\n" );
document.write( "is of the form X99Y, where Y is a positive digit.\r\n" );
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document.write( "Let's see if it can be a 2-digit number:\r\n" );
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document.write( "Then the multiplication would look like this\r\n" );
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document.write( "    999\r\n" );
document.write( "     AB\r\n" );
document.write( "   C99D\r\n" );
document.write( "  E99F \r\n" );
document.write( "  GHIJD\r\n" );
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document.write( "B cannot be 0 because the result would only amount to annexing\r\n" );
document.write( "a 0 onto one of those multiples of 999 by a single digit listed above. \r\n" );
document.write( "F cannot be 0 since A cannot be 0.  Now regardless of what positive \r\n" );
document.write( "digit F is, we will carry a 1 to the next column which would make I \r\n" );
document.write( "be 9.\r\n" );
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document.write( "So n cannot be a 2-digit number.\r\n" );
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document.write( "Let's see if n can be a 3-digit number:\r\n" );
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document.write( "    999\r\n" );
document.write( "    ABC\r\n" );
document.write( "   D99E\r\n" );
document.write( "  F99G\r\n" );
document.write( " H99I  \r\n" );
document.write( " JKLMNE\r\n" );
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document.write( "C cannot be 0 because that would amount\r\n" );
document.write( "to annexing a 0 onto a multiple of 999\r\n" );
document.write( "by a two-digit number and we have already \r\n" );
document.write( "ruled those out.\r\n" );
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document.write( "C cannot be 1, for then E would be 9.\r\n" );
document.write( "So the smallest digit we can try for C is therefore 2, so\r\n" );
document.write( "then we would have:\r\n" );
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document.write( "    999\r\n" );
document.write( "    AB2\r\n" );
document.write( "   1998\r\n" );
document.write( "  F99G\r\n" );
document.write( " H99I  \r\n" );
document.write( " JKLMN8\r\n" );
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document.write( "So far so good.  B can't be 0 for that would make G be 0,\r\n" );
document.write( "and therefore N would be 9.  So the smallest digit we can try\r\n" );
document.write( "for B is 1. That gives:\r\n" );
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document.write( "    999\r\n" );
document.write( "    A12\r\n" );
document.write( "   1998\r\n" );
document.write( "   999\r\n" );
document.write( " H99I  \r\n" );
document.write( " JKLM88\r\n" );
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document.write( "So far so good.  The smallest digit we can try\r\n" );
document.write( "for A is also 1.  That gives:\r\n" );
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document.write( "    999\r\n" );
document.write( "    112\r\n" );
document.write( "   1998\r\n" );
document.write( "   999\r\n" );
document.write( "  999  \r\n" );
document.write( " 111888\r\n" );
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document.write( "Eureka! That's it! So the smallest possible value \r\n" );
document.write( "for n is n=112.\r\n" );
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document.write( "Edwin
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