Question 127
Let $n$ be a positive integer. For each positive integer $k \le n$, let $r_k$ denote the remainder $2^n$ leaves upon division by $k$. Prove that
$$
\sum_{k=1}^{n} r_k \gt \frac{n}{2} \left( \log_2 \frac{n}{3} - 2 \right).
$$
$$
\sum_{k=1}^{n} r_k \gt \frac{n}{2} \left( \log_2 \frac{n}{3} - 2 \right).
$$