# Pow (Broken Eggs)

A farmer is taking her eggs to the market when her cart tips over shattering all of the eggs. She goes to an insurance agent unsure of how many there actually were. She needs to know this to tell the agent though. However, she does know that when she put the eggs in groups of two, there would be one left over. This also seemed to be true for groups of three, four, five, and six. But when she put them in groups of seven, there were an equal number of eggs in each group with none left over.

At first glance, some people might assume the answer is 49 because from simply looking at the problem you would say “Oh, 7 goes evenly into 49, with nothing left over” and from quick thinking you would assume the other numbers would too. But when you try and do the math, you realize that all of the numbers need to go into 48 with 1 left over, and most do, but 5 does not go evenly go into 48 with one left over. So the answer could not possibly be 49.

I began this problem not really sure how to start. I knew that the answer would have to be a multiple of seven so I went from there. Starting from 49 (because I thought it wouldn’t be 49 or lower), I tried each multiple of seven until I got to 140. This is when I started to rethink my strategy, I knew there had to be a more logical way of looking at this. I talked to my friend Michelle who made it clear there was a simpler way of looking at this. She had found the answer by laying out what she knew, and had figured it out from there. So I decided to do the same. I listed everything that I knew about the answer:

The answer had to be a multiple of 7, but 2-6 also had to go into the number before the multiple of 7, so that there would be one left over. So the multiple of seven could not be an even number because it had to have something left over. So I now know that the number is an odd multiple of seven, and I just looked at all of the multiples (not paying attention to the even multiples of 7). I...

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