document.write( "Question 357851: A woman with a basket of eggs finds that if she removes
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Algebra.Com's Answer #255492 by Edwin McCravy(20055)\"\" \"About 
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A woman with a basket of eggs finds that if she removes
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\n" ); document.write( "how many eggs does she have? Explain your reasoning.
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document.write( "Suppose she has N eggs\r\n" );
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document.write( ">>...the eggs from the basket 3 or 5 at a time, there is always 1\r\n" );
document.write( "egg left...<<\r\n" );
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document.write( "That means that N is 1 more than a multiple of both 3 and 5, which means\r\n" );
document.write( "that it is 1 more than a multiple of 15.  So we can write N as 15p + 1, where k\r\n" );
document.write( "is a positive integer.\r\n" );
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document.write( ">>...However, if she removes the eggs 7 at a time, there are no eggs left...<<\r\n" );
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document.write( "So we have to find a positive integer of the form 15p + 1 that is a multiple of 7.\r\n" );
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document.write( "Try p=1.  15p + 1 = 15(1) + 1 = 16, not a multiple of 7.\r\n" );
document.write( "Try p=2.  15p + 1 = 15(2) + 1 = 31, not a multiple of 7.\r\n" );
document.write( "Try p=3.  15p + 1 = 15(3) + 1 = 46, not a multiple of 7.\r\n" );
document.write( "Try p=4.  15p + 1 = 15(4) + 1 = 61, not a multiple of 7.\r\n" );
document.write( "Try p=5.  15p + 1 = 15(5) + 1 = 76, not a multiple of 7.\r\n" );
document.write( "Try p=6.  15p + 1 = 15(6) + 1 = 91, which is a multiple of 7, so that's the answer.\r\n" );
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document.write( "91 eggs.  You can also do it by solving the Diopantine equation\r\n" );
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document.write( "                  15p + 1 = 7q  \r\n" );
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document.write( "Are you supposed to do it that way?  That would depend on what \r\n" );
document.write( "course of mathematics you are taking.\r\n" );
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document.write( "91 = 30*3+1 = 28*5+1 = 13*7\r\n" );
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document.write( "So \r\n" );
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document.write( "when she takes out 30 groups of 3 eggs each, she takes out 90 eggs, and has 1 egg left,\r\n" );
document.write( "when she takes out 18 groups of 5 eggs each, she takes out 90 eggs, and has 1 egg left, \r\n" );
document.write( "when she takes out 13 groups of 7 eggs each, she takes out all 91 eggs and has none left.\r\n" );
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document.write( "Edwin
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