ILL 1983 Problem 56

Consider the expansion $$(1 + x + x^2 + x^3 + x^4)^{496} = a_0 + a_1x + \cdots + a_{1984}x^{1984}.$$(a) Determine the greatest common divisor of the coefficients $a_3, a_8, a_{13}, \ldots , a_{1983}.$ (b) Prove that $10^{340 }< a_{992} < 10^{347}.$