Selected problem: Broken Eggs

The Broken Egg problem states that a farmer is taking her eggs to the market but she hits a pothole and every egg gets broken. She is trying to figure out the amount of eggs that she had before they all were broken. There are certain things that she remembers. She knows that when she puts the eggs in groups of 2, 3, 4, 5, or 6 she has one egg left over as a remainder every time. However, when she put her eggs into a group of 7, there was no eggs left over. Our task is to find possible numbers that satisfy all of our constraints. I decided to create a chart of possible numbers of eggs to see if I could eventually find a combination that works.

Here is a picture of my work:





I was able to come up with viable combinations the more that I tested out numbers because I could recall some patterns of numbers relating to divisibility:

1. When numbers are divisible by 2, the last digit in the resulting number MUST be EVEN. With one remainder left over, this means that all combinations that must work has to end in an odd number.
2. When numbers are divisible by 3, you can add up every number in the digit the resulting sum is also divisible by 3. Because there has to be a remainder of 1, I needed to find numbers that were divisible by 3 and then simply add 1. For example; 301 was the first possibility that I found. 300-1=300. 3+0+0=3, and 3 is of course divisible by 3.
3. When a number is divisible by 5, the last digit in the number must be a 0 or a 5. Knowing that there has to be a remainder of 1, the digits must end in 6 or 1.

The constraints that I knew allowed me to only guess and check certain numbers, while eliminating a lot of other possibilities. Overall, I was able to come up with 4 viable solutions to the amount of eggs that the farmer might have had; 301,721,2401, or 4501. It is helpful to notice that each number that was found has a last digit of 1.