Proof Writing & Problem Solving · Challenge problem

The proof that $H_n = 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1n$ is never an integer for $n \ge 2$ focuses on the largest power of $2$ that is at most $n$. Checking the argument for $n = 10$, write $H_{10} = \dfrac{p}{q}$ in lowest terms. What is the largest $k$ such that $2^k$ divides $q$?