Solve the equation $\sigma{(n)}=n$

Question

Find all positive $n$ such that $$\sigma{(n)}=n$$ where $\sigma{(n)}$ is the sum-of-divisors function.

We write this equation as following: $$\dfrac{\sigma{(n)}}{n}=\sum_{d|n}\dfrac{1}{d}=1$$ I checked that the number $n=1$ is one solution, but are there others?

Answer 1

Hint: $n$ is a divisor of $n$. What happens if $n$ has any other divisors?

Answer 2

If $n\neq 1$, then $n$ and $1$ would be different divisors of $n$, so the sum of divisors would be at least $n+1.$ Hence, only solution is $n=1$.