# If $n>$1 and $S_n= \frac12 + \frac13 +\cdots + \frac1n$, show $S_n$ is not an integer. [duplicate]

At the back of the book which contains this problem, a hint is given to consider $$S_n\cdot2^{k-1}\cdot3\cdot5\cdot9\cdot\ldots$$ where $2^k \le n < 2^{k+1}$.

I don't know how this helps? I know that every positive integer can be written as a sum of powers of 2, but not sure if this is relevant.

## marked as duplicate by user7530, Ian Coley, Pedro Tamaroff♦, Antonio Vargas, user127096Mar 19 '14 at 19:15

• An answer here : math.stackexchange.com/questions/2746/… – user119228 Mar 19 '14 at 18:27
• I suspect the hint meant to have a $7$ instead of that $9$, i.e., multiply $S_n$ by a power of $2$ (that turns out to leave a single $2$ in the denominator of the product) and all odd numbers up to $n$. – Barry Cipra Mar 19 '14 at 18:49