Let $n$ be a positive integer. Navinim writes down all positive square numbers that divide $n$ on a blackboard. For each number $k$ on the blackboard, Navagem replaces it with $d(k)$. Show that the sum of all numbers on the blackboard now is a perfect square. (Note: $d(k)$ denotes the number of divisors of $k$.) (Proposed by Ivan Chan Guan Yu)