Consider the sixteen cells in a $4\times4$ square. Two (distinct) cells are neighbors, if they share a horizontal or vertical edge; note that every cell has two or three or four neighbors. Now each of the sixteen cells contains an integer, so that for every cell the integers in the neighboring cells add up to the sum $120$.
(a) What are the possible values for the sum of the sixteen integers?
(b) Does there exist such a $4\times4$ square in which the sixteen integers are pairwise distinct?