Let $d_1,d_2,\ldots,d_k,$ be all the factors of a positive integer $n,$ including $1,$ and $n.$ Suppose $d_1+d_2+\ldots+d_k=72.$ Then, find the value of $\frac{1}{d_1}+\frac{1}{d_2}+\ldots+\frac{1}{d_k}.$
My Attempt:
Let $S=\frac{1}{d_1}+\frac{1}{d_2}+\ldots+\frac{1}{d_k}.$ Then, $S=\frac{1}{n}[\frac{n}{d_1}+\frac{n}{d_2}+\ldots+\frac{n}{d_k}].$ But, each $\frac{n}{d_i},i\in\{1,2,\ldots,k\}$ is also a factor of $n$. Since we have $k$ such terms, we can say that $\frac{n}{d_1}+\frac{n}{d_2}+\ldots+\frac{n}{d_k}=72.$ Hence, $S=\frac{72}{n}.$
My Question:
Is the answer and/or method right? Are there any other (possibly more informative) ways to approach this question?