JEE Novice - (5)

The second equation can be written as

$(a + 1)(b + 1) = 323 = 17 \times 19,$

and the third equation can be written as

$(b + 1)(c + 1) = 399 = 3 \times 7 \times 19.$

By the uniqueness of prime factorization, (i.e., the Fundamental Theorem of Arithmetic), since $(b + 1)$ is a common factor of these two equations we must have that $(b + 1) = 19 \Longrightarrow b = 18.$

This implies that $(a + 1) = 17 \Longrightarrow a = 16$ and $(c + 1) = 21 \Longrightarrow c = 20.$

So $abc = 16 \times 18 \times 20 = 2 \times 8 \times 3 \times 6 \times 4 \times 5 = \dfrac{8!}{7},$ and so for $abcd$ to equal $8!$ we will require that $d = \boxed{7}.$