smc

Question

For every composite positive integer n, define r(n) to be the sum of the factors in the prime factorization of n. For example, r(50) = 12 because the prime factorization of 50 is 2 5^2, and 2 + 5 + 5 = 12. What is the range of the function r, r(n): n is a composite positive integer ?

Accepted Answer

This problem becomes simple once we recognize that the domain of the function is 4, 6, 8, 9, 10, 12, 14, 15, . By evaluating r(4) to be 4, we can see that (E) is incorrect. Evaluating r(6) to be 5, we see that both (B) and (C) are incorrect. Since our domain consists of composite numbers, which, by definition, are a product of at least two positive primes, the minimum value of r(n) is 4, so (A) is incorrect. That leaves us with the set of integers greater than 3.