Answer

The smallest possible number of eggs that Mrs. Covey could have taken to market is 719. 

After selling half the number and giving half an egg over she would have 359 left; after the second transaction she would have 239 left; after the third deal, 179; and after the fourth, 143. 

This last number she could divide equally among her thirteen friends, giving each 11, and she would not have broken an egg.