nt.number theory - Smallest integer not divisible by integers in a finite set

Hello all, if $a_1,a_2, ldots a_t$ are $t$ integers $geq 2$, the set
$G(a_1,a_2, ldots a_t)=lbrace N geq 1 |$ In any sequence of $N$ consecutive
integers there is at least one not divisible by any of $a_1,a_2, ldots a_trbrace$
is nonempty (it contains $a_1a_2 ldots a_t$) so it has a minimal element
which we denote by $g(a_1,a_2, ldots a_t)$.

Question 1 : Is there a uniform bound $gamma (t)$, depending
only on $t$, such that $gamma (t) geq g(a_1,a_2, ldots a_t)$ for any
$a_1,a_2, ldots a_t$ ? For example, we may take $gamma(2)=4$.

Question 2 : If $gamma$ is well-defined,
are any asymptotics known about $gamma(t)$ ?