2023 AMC 10A Problems/Problem 23

If the positive integer c has positive integer divisors a and b with c = ab, then a and b are said to be $\textit{complementary}$ divisors of c. Suppose that N is a positive integer that has one complementary pair of divisors that differ by 20 and another pair of complementary divisors that differ by 23. What is the sum of the digits of N?

$\textbf{(A) } 9 \qquad \textbf{(B) } 13\qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$

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2023 AMC 10A Problems/Problem 23