Let $t$ be TNYWR.

Determine the number of integers $b>0$ for which $30 t$ is divisible by $b!$.

(If $n$ is a positive integer, the symbol $n$ ! (read " $n$ factorial") represents the product of the integers from 1 to $n$. For example, $4!=(1)(2)(3)(4)$ or $4!=24$.)
We do some preliminary work assuming that the number $t$ to be received is a positive integer. (This turns out to be the case; if it were not the case, we might have to change strategies.)

Since $1!=1$, then $30 t$ is divisible by 1 ! regardless of the value of $t$.

Since $2!=2$, then $30 t=2(15 t)$ is divisible by 2 ! regardless of the value of $t$.

Since $3!=6$, then $30 t=6(15 t)$ is divisible by 3 ! regardless of the value of $t$.

Next, consider $4!=24$.

Since $\frac{30 t}{4!}=\frac{30 t}{24}=\frac{5 t}{4}$, then $30 t$ is divisible by 4 ! if and only if $t$ is divisible by 4 .

Next consider $5!=120$.

Since $\frac{30 t}{5!}=\frac{30 t}{120}=\frac{t}{4}$, then $30 t$ is divisible by 5 ! if and only if $t$ is divisible by 4 .

Next consider $6!=720$.

Since $\frac{30 t}{6!}=\frac{30 t}{720}=\frac{t}{24}$, then $30 t$ is divisible by $6!$ if and only if $t$ is divisible by 24 .

Since the answer to (b) is 56 , then $t=56$.

Since $t$ is divisible by 4 but not by 24 , then $30 t$ is divisible by 1 !, 2 !, 3 !, 4 !, and 5 !, but not by 6 !.

Furthermore, $30 t=1680$ and if $n \geq 7$, then $n!\geq 7!=5040$, so $30 t$ is not divisible by $n!$ for $n \geq 7$ (because $n!>30 t$ ).

Therefore, there are 5 integers $b$ for which $30 t$ is divisible by $b$ !.

ANSWER: $52,56,5$