document.write( "Question 331715: Hi there, I am stumped at how to do about this. Can someone please help me? Thank you!\r
\n" ); document.write( "\n" ); document.write( "A woman with a basket of eggs finds that if she removes
\n" ); document.write( "the eggs from the basket 3 or 5 at a time, there is always 1
\n" ); document.write( "egg left. However, if she removes the eggs 7 at a time,
\n" ); document.write( "there are no eggs left. If the basket holds up to 100 eggs,
\n" ); document.write( "how many eggs does she have? Explain your reasoning. \n" ); document.write( "

Algebra.Com's Answer #237792 by Edwin McCravy(20056) $\"\"$ $\"About$

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Hi there, I am stumped at how to do about this. Can someone please help me? Thank you!
\n" ); document.write( "A woman with a basket of eggs finds that if she removes
\n" ); document.write( "the eggs from the basket 3 or 5 at a time, there is always 1
\n" ); document.write( "egg left. However, if she removes the eggs 7 at a time,
\n" ); document.write( "there are no eggs left. If the basket holds up to 100 eggs,
\n" ); document.write( "how many eggs does she have? Explain your reasoning.
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document.write( "Suppose she has n eggs.  Then n-1 is both a multiple of 3 and\r\n" );
document.write( "5 which means that n-1 is a multiple of 15. if she removes the \r\n" );
document.write( "eggs 7 at a time, there are no eggs left. That means n is a \r\n" );
document.write( "multiple of 7.\r\n" );
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document.write( "Since n-1 is a multiple of 15, there exists positive integer p \r\n" );
document.write( "so that n-1 = 15p\r\n" );
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document.write( "Since n is a multiple of 7, there exists positive integer q \r\n" );
document.write( "so that n = 7q.\r\n" );
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document.write( "So we have this system of equations:\r\n" );
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document.write( "So we substitute 7q for n from the second equation, into the first equation\r\n" );
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document.write( "The smallest coefficient is 7, and it has absolute value 7,\r\n" );
document.write( "so write all numbers in terms of their nearest multiple of 7.\r\n" );
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document.write( "Divide every term through by 7\r\n" );
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document.write( "Isolate the fraction terms:\r\n" );
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document.write( "The left side is an integer and so is the right side.\r\n" );
document.write( "Let that integer be A\r\n" );
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document.write( "then we have this system:\r\n" );
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document.write( "Clear the second of fractions by multiplying through by 7\r\n" );
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document.write( "Solve the second one for p\r\n" );
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document.write( "Substitute 7A-1 for p in the other equation\r\n" );
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document.write( "Substitute that in\r\n" );
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document.write( "Therefore A = 1 because 1 is the only integer\r\n" );
document.write( "between those two values.\r\n" );
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document.write( " Since \r\n" );
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document.write( "  \r\n" );
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document.write( "Therefore she has 91 eggs in her basket.\r\n" );
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document.write( "A woman with a basket of 91 eggs finds that if she removes\r\n" );
document.write( "the eggs from the basket 3 at a time as we can see from the\r\n" );
document.write( "first two divisions below that after she removes 30 groups \r\n" );
document.write( "of 3 or 18 groups of 5 she will in both cases have 1 remaining \r\n" );
document.write( "because there is a 1 remainder in the first two cases.  The \r\n" );
document.write( "third division below shows that when she removes 13 groups of 7 \r\n" );
document.write( "each she has none remaining because there is a 0 remainder.\r\n" );
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document.write( "  30         18       13\r\n" );
document.write( "3)91       5)91     7)91\r\n" );
document.write( "  90         5        7\r\n" );
document.write( "   1         41       21\r\n" );
document.write( "             40       21\r\n" );
document.write( "              1        0\r\n" );
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document.write( "Edwin

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