Answer: EGG = 288, JAM = 576 (A = 7, E = 2, G = 8, J = 5, M = 6)
 
 
E$\times$E is less than 10, so E is 1, 2 or 3

EGG$\times$1 = EGG not JAM so E is 2 or 3


 A and M are different so G$\times$2 or G$\times$3 is 10 or more.

But for E=3, J=3$\times$3=9 maximum, so there shouldn't be a carried digit, so E can't be 3


2$\times$G is at most 18 so the digit which is carried is 1, not 2

 Only G = 8, M = 6, A = 7 has no repeats