document.write( "Question 1169153: Let n​,​7874185de be a​ base-ten numeral with d and e its last two digits. Give all of the choices of the​ two-digit numbers de for which n is divisible by 12. \n" ); document.write( "
Algebra.Com's Answer #794030 by Edwin McCravy(20064)\"\" \"About 
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document.write( "Since \"de\" = 10d + e,\r\n" );
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document.write( "\"n+=+7874185de+=+787418500+%2B+10d+%2B+e\"\r\n" );
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document.write( "\"n%2F12=65618208%2B1%2F3%2B%2810d%2Be%29%2F12\"\r\n" );
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document.write( "So \"1%2F3%2B%2810d%2Be%29%2F12\" must be a non-negative integer, which we can represent by the letter A\r\n" );
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document.write( "\"1%2F3%2B%2810d%2Be%29%2F12=A\"\r\n" );
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document.write( "Multiply through by 12 to clear of fractions:\r\n" );
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document.write( "\"12%2A%281%2F3%29%2B12%2A%28%2810d%2Be%29%2F12%29=12%2AA\"\r\n" );
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document.write( "\"4%2B10d%2Be=12A\"\r\n" );
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document.write( "\"10d%2Be=12A-4\"\r\n" );
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document.write( "Since de = 10d+e must be a 2-digit number, so must 12A-4\r\n" );
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document.write( "The smallest 2-digit number is 10 and the largest 2-digit number is 99, so\r\n" );
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document.write( "\"10%3C=12A-4%3C=99\"\r\n" );
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document.write( "Add 4 to all three sides\r\n" );
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document.write( "\"14%3C=12A%3C=103\"\r\n" );
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document.write( "Divide through by 12\r\n" );
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document.write( "\"14%2F12%3C=12A%2F12%3C=103%2F12\"\r\n" );
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document.write( "\"1%261%2F6%3C=A%3C=8%267%2F12\"\r\n" );
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document.write( "Since A must be a non-negative integer,\r\n" );
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document.write( "\"2%3C=A%3C=8\"\r\n" );
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document.write( "So there are 7 choices for A, 2 through 8, inclusive\r\n" );
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document.write( "\"10d%2Be=12A-4\"\r\n" );
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document.write( "A  12A-4  de       n = 7874185de (to check) \r\n" );
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document.write( "2   20    20           787418520 = (12)(65618210) \r\n" );
document.write( "3   32    32           787418532 = (12)(65618211)\r\n" );
document.write( "4   44    44           787418544 = (12)(65618212)\r\n" );
document.write( "5   56    56           787418556 = (12)(65618213)\r\n" );
document.write( "6   68    68           787418568 = (12)(65618214)\r\n" );
document.write( "7   80    80           787418580 = (12)(65618215)\r\n" );
document.write( "8   92    92           787418592 = (12)(65618216)\r\n" );
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document.write( "The 7 choices for de are in red.  \r\n" );
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document.write( "Edwin
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