If the positive integer $c$ has positive integer divisors $a$ and $b$ with $c=ab$, then $a$ and $b$ are said to be complementary divisors of $c$. Suppose that $N$ is a positive integer that has one complementary pair of divisors that differ by $10$ and another pair of complementary divisors that differ by $13$. What is the sum of the digits of $N$?