I am working on a problem.
Consider the set S of integers $1,2,...,n$. Let $2^k$ be the integer in S that is the highest power of $2$. Prove that $2^k$ is not a divisor of any other integer in S. Hence prove that $ \sum_{j=1}^{n} \frac{1}{j}$ is not an integer if $n>1$.
I turned out with the first one, but still got stuck to prove the sum is not an integer. I don't know how to use the first part to prove the second part.
Any hint is appreciated.
Each of the numbers from $1$ to $n$ can be written in the form $m2^s,$ where $m$ is odd. By definition of $k, s \le k$ in every case, and in all but one case, $s \le k-1.$ Now, suppose the sum is equal to an integer, and multiply both sides by $2^{k-1}.$ One of the terms in the equation is a fraction with an even denominator, and all the rest have odd denominators. Deduce a contraction.