Show that sum of divisors of a composite number $n$ is $> n+ \sqrt{n}$

Question

The hint says to use : When $1\lt d\lt n$, $1 \lt n/d \lt n$. If $d\le \sqrt{n}$, then $n/d \ge \sqrt{n}$

My try :

Since there will be atleast half divisors $> \sqrt{n}$, the sum of divisors will be atleast $\dfrac{\tau (n)}{2}\sqrt{n}$ Not sure how to proceed further. Any help ?

Answer 1

Use the hint to prove that there exists a divisor $\sqrt{n} \leq d <n$.

Then $n$ and $d$ are divisors of $n$ and their sum is....

Answer 2

n is divisor of n , and there is atleast one $d >= \sqrt n$