Question 150876
You need to think this one through before trying some potential solutions
1) You need to realize that the answer must not be a number that is divisible by 2,3,4,5,6 -- so you immediately start thinking "prime numbers" greater than 7 are good candidates to try
2) In order for the 'solution' to leave a remainder of 1 when divided by 5, the solution must end in either a 1 or a 6.
3) A solution that ends in 6 won't work, since that solution fails when dividing by 2

So we need to multiply 7 by a number that results in a solution that ends in a 1. The only was that can happen is if the number we multiply by ends in a 3. (since 3 * 7 ends in a 1)

So, we are looking for a prime number that ends in 3.

Now comes the trial and error
try 13, 23, 43, 53 ...
13*7 = 91
works for 2, works for 3. fails for 4 since there are 3 eggs left
try 23
23*7 = 161
works for 2, fails for 3 with 2 eggs left
try 43
43*7 = 301
works for 2, 3, 4, 5, 6  tada!