document.write( "Question 651783: I have had this problem for a month and I can't figure out what to do.
\n" ); document.write( "Maybe you can help.\r
\n" ); document.write( "\n" ); document.write( "A farmer is taking her eggs to the market in a cart, but she hits a
\n" ); document.write( "pothole, which knocks over all the containers of eggs. Though she is
\n" ); document.write( "unhurt, every egg is broken. So she goes to her insurance agent, who
\n" ); document.write( "asks her how many eggs she had. She says she doesn't know, but she
\n" ); document.write( "remembers somethings from various ways she tried packing the eggs.\r
\n" ); document.write( "\n" ); document.write( "When she put the eggs in groups of two, three, four, five, and six
\n" ); document.write( "there was one egg left over, but when she put them in groups of seven
\n" ); document.write( "they ended up in complete groups with no eggs left over.\r
\n" ); document.write( "\n" ); document.write( "What can the farmer figure from this information about the number of
\n" ); document.write( "eggs she had? Is there more than one answer?\r
\n" ); document.write( "\n" ); document.write( "I have already tried multiples of seven, but I can't figure the
\n" ); document.write( "answer out. Could you please tell me how to get the answer and also
\n" ); document.write( "tell me what it is?\r
\n" ); document.write( "\n" ); document.write( "Thanks for your help.
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Algebra.Com's Answer #407805 by MathLover1(20850) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "If there were $\"no\"$ eggs $\"left\"$ $\"+over\"$ when put into groups of $\"seven\"$ there $\"must\"$ have been a $\"multiple\"$ of $\"7\"$ eggs. \r
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\n" ); document.write( "\n" ); document.write( "We need to find $\"n\"$ where
\n" ); document.write( " $\"n+=+7a\"$
\n" ); document.write( " $\"n+=+6b+%2B+1\"$
\n" ); document.write( " $\"n+=+5c+%2B+1\"$
\n" ); document.write( " $\"n+=+4d+%2B+1\"$ \r
\n" ); document.write( "\n" ); document.write( "where $\"a\"$ , $\"b\"$ , $\"c\"$ , and $\"+d\"$ are the number of $\"full\"$ $\"groups\"$ when the eggs are put into groups of $\"7\"$ , $\"6\"$ , $\"5\"$ , and $\"4\"$ .\r
\n" ); document.write( "\n" ); document.write( " $\"If\"$ there is $\"one\"$ $\"+egg\"$ $\"+left\"$ $\"+over\"$ when the eggs are put into groups of $\"6\"$ there will automatically be $\"one\"$ $\"+egg\"$ $\"+left\"$ $\"+over\"$ when they are put into groups of $\"2\"$ or groups of $\"3\"$ . So we can forget groups of $\"2\"$ and $\"3\"$ .\r
\n" ); document.write( "\n" ); document.write( "The $\"smallest\"$ number which is $\"divisible\"$ by $\"5\"$ and $\"6\"$ is $\"30\"$ , but $\"30\"$ isn't divisible by $\"4\"$ , so the smallest number which will divide by $\"4\"$ , $\"5\"$ , and $\"6\"$ is $\"60\"$ .\r
\n" ); document.write( "\n" ); document.write( "So we need\r
\n" ); document.write( "\n" ); document.write( " $\"n+=+60e+%2B+1\"$
\n" ); document.write( "but also\r
\n" ); document.write( "\n" ); document.write( " $\"n+=+7a\"$ \r
\n" ); document.write( "\n" ); document.write( "The easiest way to solve this is to try values $\"1\"$ , $\"+2\"$ , $\"3\"$ , $\"+4\"$ , $\"5\"$ ,... for $\"e\"$ to get values of $\"61\"$ , $\"121\"$ , $\"181\"$ , $\"241\"$ , $\"301%7D%7D%2C%7B%7B%7B420\"$ ... for $\"n\"$ , and to pick the first of these which divides by $\"7\"$ . This is $\"301\"$ .\r
\n" ); document.write( "\n" ); document.write( "So the $\"smallest\"$ number of eggs is $\"301\"$ .\r
\n" ); document.write( "\n" ); document.write( "Adding $\"60+%2A+7\"$ to a valid solution will also give a valid solution.\r
\n" ); document.write( "\n" ); document.write( "So the next solution is $\"301+%2B+420+=+721\"$ .\r
\n" ); document.write( "\n" ); document.write( "All the solutions are $\"301\"$ , $\"721\"$ , $\"1141\"$ , $\"1561\"$ , $\"1981\"$ , ... (i.e. add $\"420\"$ each time)\r
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