document.write( "Question 1205868: Hi, So my question is:
\n" ); document.write( "A bag of lollies is shared. When shared equally among 4 people, there are 2 lollies left. When shared equally among 9 people, there are 5 lollies left. When shared equally among 7 people, there are 2 lollies left.
\n" ); document.write( "Determine the smallest possible number of lollies that could have been in the bag as well as the general solution.
\n" ); document.write( "I have tried a few different ways and get different answers.I either get 86 or 94. Can you help? Thank you \n" ); document.write( "

Algebra.Com's Answer #842955 by mccravyedwin(407) $\"\"$ $\"About$

You can put this solution on YOUR website!
Question 1205868
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document.write( "Ikleyn believes trial and error is legitimate, but it isn't! There is no\r\n" );
document.write( "guarantee that only a small number of trials is all you'll need.  You may have\r\n" );
document.write( "to try hundreds of values.  A problem is not solved mathematically unless it\r\n" );
document.write( "requires no trial and error.  The way I solved it above, \"long and scary\" as she\r\n" );
document.write( "thinks it is, (longer than necessary because I don't skip steps), but the way I\r\n" );
document.write( "solved it requires no trial and error.\r\n" );
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document.write( "Many mathematicians have balked at the computer proof of the four-color problem.\r\n" );
document.write( "For the only way it has been \"proved\" is not with mathematics, but to check out\r\n" );
document.write( "all the possible types of maps to show that all types can be colored with only 4\r\n" );
document.write( "colors and no two adjacent territories are colored the same. There are so many\r\n" );
document.write( "types of maps to try, that only a computer could try them all.  Mathematicians\r\n" );
document.write( "are still looking for a mathematical proof of the problem. \r\n" );
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document.write( "Above I started this way.\r\n" );
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document.write( "Let n be the smallest possible answer.\r\n" );
document.write( "Then there are positive integers p,q,r such that\r\n" );
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document.write( " with k positive, and small as possible\r\n" );
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document.write( "Sure, when I got here, I could have stopped and started trying values for k,\r\n" );
document.write( "but there is no guarantee that, as Ikleyn stated, \r\n" );
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document.write( "\"it will not take long time\". \r\n" );
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document.write( "before I found one that worked. I could have done as she did:\r\n" );
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document.write( "\"Try some values k= 1, 2, 3, 4, 5 . . . .It will not take long time...\"\r\n" );
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document.write( "So I could have tried k=3, then instantly p=21, r=12,\r\n" );
document.write( "4p+2=n \r\n" );
document.write( "4(21)+2 = 86\r\n" );
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document.write( "And that would be the end. But it's sheer luck that that's all it took.\r\n" );
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document.write( "What is interesting is the fact that this:\r\n" );
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document.write( "\"When shared equally among 9 people, there are 5 lollies left.\"\r\n" );
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document.write( "was redundant information. \r\n" );
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document.write( "The problem would have much harder if it had been stated this way:\r\n" );
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document.write( "A bag of lollies is shared. When shared equally among 4 people, there are 2 \r\n" );
document.write( "lollies left. When shared equally among 9 people, there are 5 lollies left. \r\n" );
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document.write( "Edwin

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