Question 1044685
We can brute-force our way to a solution:

1 modulo 2 ===> 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29, 31
1 modulo 3 ===> 1,4,7,10,13,16,19,22,25,28,31
1 modulo 5 ===> 1,6,11,16,21,26,31

Barring the possibility that there is only one egg, the least number of eggs for this grouping to be done is 31.

(There is a conventional method of doing this problem, by means of the chinese remainder theorem, but the 
solution above should suffice.)