Question:
If $n$ and $k$ are positive integers such that $5<\frac nk<6$, then what is the smallest possible value of $\frac{\mathop{\text{lcm}}[n,k]}{\gcd(n,k)}$?

Answer:
We can consider both $n$ and $k$ as multiples of their greatest common divisor: \begin{align*}
n &= n'\cdot\gcd(n,k), \\
k &= k'\cdot\gcd(n,k),
\end{align*}where $n'$ and $k'$ are relatively prime integers. Then $\mathop{\text{lcm}}[n,k] = \frac{n\cdot k}{\gcd(n,k)} = n'\cdot k'\cdot\gcd(n,k)$, so $$\frac{\mathop{\text{lcm}}[n,k]}{\gcd(n,k)} = n'k'.$$We have $\frac{n'}{k'} = \frac nk$. So, we wish to minimize $n'k'$ under the constraint that $5<\frac{n'}{k'}<6$. That is, we wish to find the smallest possible product of the numerator and denominator of a fraction whose value is between 5 and 6. Clearly the denominator $k'$ is at least $2$, and the numerator $n'$ is at least $5(2)+1=11$, so the smallest possible value for $n'k'$ is $(11)(2)=\boxed{22}$.

Note that this result, $\frac{\mathop{\text{lcm}}[n,k]}{\gcd(n,k)}=22$, can be achieved by the example $n=11,k=2$.