jmc

$$\text{gcd}(7560,8400)=840=2^3\cdot 3^1\cdot 5^1\cdot 7^1$$The common divisors of 7560 and 8400 are the divisors of their GCD: $$t(840)=(3+1)(1+1)(1+1)(1+1)=\boxed{32}.$$