document.write( "Question 836189: if the eggs in a basket are removed 2 at a time, one eggs will remain. If they are removed 3 at a time, 2 eggs remain. If the eggs are removed 4,5 or 6 at a time, then 3,4 and 5 eggs remian, respectively. But if they are taken out 7 at a time, no eggs will be left over. Find the least numbe of eggs that could be in the basket. \n" ); document.write( "
Algebra.Com's Answer #504035 by Edwin McCravy(20056)\"\" \"About 
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document.write( "Let the number be N\r\n" );
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\n" ); document.write( "if the eggs in a basket are removed 2 at a time, one eggs will remain.
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document.write( "So N is 1 less than a multiple of 2, so N=2A-1 for some positive integer A.\r\n" );
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\n" ); document.write( "If they are removed 3 at a time, 2 eggs remain.
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document.write( "So N is 2 more, and therefore 1 less than, a multiple of 3, so N=3B-1 for some positive integer B.\r\n" );
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\n" ); document.write( "If the eggs are removed 4...at a time, then 3...eggs remain,
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document.write( "So N is 3 more, and therefore 1 less than, a multiple of 4, so N=4C-1 for some positive integer C\r\n" );
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\n" ); document.write( "If the eggs are removed ...5... at a time, then ...4... eggs remain,
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document.write( "So N is 4 more, and therefore 1 less than, a multiple of 5, so N=5D-1 for some positive integer D.\r\n" );
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\n" ); document.write( "If the eggs are removed ...6 at a time, then ...5 eggs remain,
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document.write( "So N is 5 more, and therefore 1 less than, a multiple of 6, so N=6E-1 for some positive integer E\r\n" );
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\n" ); document.write( "But if they are taken out 7 at a time, no eggs will be left over.
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document.write( "So N is a multiple of 7, so N=7F for some positive integer F\r\n" );
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document.write( "So\r\n" );
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document.write( "N = 2A-1 = 3B-1 = 4C-1 = 5D-1 = 6E-1 = 7F\r\n" );
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document.write( "Add 1 to all those:\r\n" );
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document.write( "N+1 = 2A = 3B = 4C = 5D = 6E = 7F+1\r\n" );
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document.write( "N+1=7F+1 has to be a common multiple of 2,3,4,5,6.  The least\r\n" );
document.write( "common multiple of 2,3,4,5,6 is 60.  So N+1 is a multiple of\r\n" );
document.write( "60.  So N is 1 less than a multiple of 60.\r\n" );
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document.write( "So we find the least multiple of 60 that is 1 more than a multiple\r\n" );
document.write( "of 60.\r\n" );
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document.write( "The first multiple of 60 is 60 itself.  1 less than 60 is 59, but\r\n" );
document.write( "59 is not a multiple of 7.\r\n" );
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document.write( "The next multiple of 60 is 120.  1 less than 120 is 119, and sure\r\n" );
document.write( "enough, 119 is a multiple of 7. since 119 = 17*7.\r\n" );
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document.write( "Answer: 119 \r\n" );
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document.write( "Edwin
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