If $\gcd(x,y)=1379,\operatorname{lcm}(x,y)=n!$ then find $n$ If $\gcd(x,y)=1379,\operatorname{lcm}(x,y)=n! $ then find $n$  ($x,y,n$ are positive integers).
I tried using the relation: $\gcd(x,y)\cdot\operatorname{lcm}(x,y)=xy$ and also wrote $1379=7\cdot197$ but nothing more.
Since $\gcd(x,y)=7\cdot 197$, we can assume $x=7\cdot 197a$, and $y=7\cdot 197b$, where $\gcd(a,b)=1$. Then $\text{lcm}(x,y)=7\cdot 197\cdot ab=n!$. Thus, $n\geqslant 197$. Actually, we can assign different prime factors of $\frac{n!}{7\cdot 197}$ to $a$ and $b$ respectively to make sure they are coprime.
$n$ can be any integer which is no smaller than $197$.