AMC 10A 2023 (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 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

\text{(A)}\;9

\text{(B)}\;13

\text{(C)}\;15

\text{(D)}\;17

\text{(E)}\;19