<s>[INST] If $a=7!$, $b=_{13}P_k$, $\frac{ab}{\operatorname{lcm}(a,b)}=120$, find the $k$ 
Question:
$a=7!$, $b=_{13}P_k$, $\dfrac{ab}{\operatorname{lcm}(a,b)}=120$, then find the $k$.

My attempts:
$$\frac{ab}{\operatorname{lcm}(a,b)}=\gcd{(a,b)}=120$$
$\gcd{(7!, _{13}P_k)}=120 \Longrightarrow \begin{cases} \dfrac{13!}{(13-k)!×5!}\in \mathbb {Z^+} \\ \dfrac{13!}{(13-k)!×5!} \mod 2≠0 \\
\dfrac{13!}{(13-k)!×5!} \mod 3≠0 \\
\dfrac{13!}{(13-k)!×5!} \mod 7≠0  \end{cases} \Longrightarrow \begin{cases} 
\dfrac{13×12×\cdots (13-k+1)!}{120} \in \mathbb {Z^+} \\ 
 \dfrac{13!}{(13-k)!×5!} \mod 2≠0 \\
\dfrac{13!}{(13-k)!×5!} \mod 3≠0 \\
\dfrac{13!}{(13-k)!×5!} \mod 7≠0  \end{cases} \Longrightarrow 13-k+1=10 \Longrightarrow k=4$
Is my solution correct? Do I have any missing or unnecessary/unneeded steps?
 [/INST] This was based on the original post
The problem as stated is incorrect or there is a typo. If $\frac{ab}{\gcd(a,b)}=120$, then that would mean $\text{lcm}(a,b)=120$. But $120=\text{lcm}(a,b) \geq a=7!$. This is not possible.
After the original post got modified:
Perhaps, the $\gcd(a,b)=120$. In which case we can do the following:
\begin{align*}
\gcd(a,b)&=120\\
\gcd(7!,b)&=120\\
\gcd\left(42,\frac{b}{120}\right)&=1
\end{align*}
This means $\frac{b}{120}$ is (at least) not divisible by $2,3$ and $7$, where  $b=\frac{13!}{(13-k)!}$. We know that $13!=2^{10} \cdot 5^2 \cdot 7^{1} \dotsb$. So
$$\frac{b}{120}=\frac{13!}{(2^3\cdot 3 \cdot 5) \, (13-k)!}=\frac{2^{7} \cdot 3^4\cdot 5^1 \cdot 7^{1} \dotsb}{(13-k)!}$$
This means our $k$ should be such that the prime factorization of $(13-k)!$ should also have exactly these powers of the primes $2,3$ and $7$.
Since $2$ should only appear with exponent $7$ in the prime factorization of $(13-k)!$, so $13-k \geq 8 \implies k \leq 5$ and $13-k \leq 9 \implies k \geq 4$.
Since $3$ should only appear with exponent $4$ in the prime factorization of $(13-k)!$, so $13-k \geq 9 \implies k \leq 4$.
Thus $\color{red}{k=4}$ will satisfy the gcd condition.
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