Existence of a prime in between two integers, of which the larger integer is divisible by all prime divisors of the smaller integer.

Question

Let $a$, $b \in \mathbb{N}$ and $3 < a < b$. Suppose all prime divisors of $a$ divide $b$ and all prime divisors of $b$ less than $a$ also divide $a$. Does there always exist a prime $p$ such that $a<p<b$ and $p\nmid b$.

I was trying to find a counterexample but I am not good with programming.

Answer 1

Let $a=8$, $b=10$. Then the only prime divisor of $2$ also divides $b$, but $9$ is not prime.