A natural number N is such that it can be expressed as N = p + q + r, where p, q and r are distinct factors of N. How many numbers below 50 have this property?

Writing p = N/a etc. needs 1/a + 1/b + 1/c = 1 with distinct a, b, c, whose only solution is (2, 3, 6). So N must be a multiple of 6 (N = N/2 + N/3 + N/6): 6, 12, ..., 48 = 8 numbers. (Answer as per the official UPSC answer key.)