document.write( "Question 1098659: let n be the largest integer less than 10000 that leaves remainder 1 when divided 2,3,4,5,6,7 and 8.the sum of the digits of n is \n" ); document.write( "
Algebra.Com's Answer #713027 by Edwin McCravy(20062)\"\" \"About 
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document.write( "Any integer which is 1 more than a multiple of all those\r\n" );
document.write( "will leave remainder 1 when divided by any of those.  Any \r\n" );
document.write( "integer which is a multiple of all those is a multiple of \r\n" );
document.write( "their least common multiple.\r\n" );
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document.write( "2,3,4,5,6,7,8 = 2,3,2x2,5,2x3,7,2x2x2\r\n" );
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document.write( "So the LCM must have 3 factors of 2, 1 factor of 3, 1 factor \r\n" );
document.write( "of 5, and 1 factor of 7. So the LCM = 2x2x2x3x5x7 = 840, so \r\n" );
document.write( "any number of the form 840k+1 will leave remainder 1 when \r\n" );
document.write( "divided by 2,3,4,5,6,7 or 8.\r\n" );
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document.write( "We want the largest such integer less than 10000\r\n" );
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document.write( "840k+1 < 10000\r\n" );
document.write( "  840k < 9999\r\n" );
document.write( "     k < 9999/840\r\n" );
document.write( "     k < 11.90357143...\r\n" );
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document.write( "So the largest integer k can be is 11\r\n" );
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document.write( "840k+1 = 840(11)+1 = 9241 = n\r\n" );
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document.write( "The sum of the digits of n is 9+2+4+1 = 16\r\n" );
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document.write( "Edwin
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