How many possible values of (p + q + r) are there satisfying (1)/(p) + (1)/(q) + (1)/(r) = 1 , where p, q and r are natural numbers (not necessarily distinct)?

To solve the equation 1/p + 1/q + 1/r = 1 where p, q, and r are natural numbers, we can assume p is less than or equal to q, and q is less than or equal to r. If p equals 1, the sum would exceed 1, so p must be at least 2. Case 1: If p = 2 The equation becomes 1/q + 1/r = 1/2. The possible pairs for (q, r) are (3, 6), (4, 4), and (2, infinity is not possible). - For (2, 3, 6), the sum p + q + r is 2 + 3 + 6 = 11. - For (2, 4, 4), the sum p + q + r is 2 + 4 + 4 = 10. Case 2: If p = 3 The equation becomes 1/q + 1/r = 2/3. The only possible pair for (q, r) where q is at least 3 is (3, 3). - For (3, 3, 3), the sum p + q + r is 3 + 3 + 3 = 9. If p is greater than 3, the sum 1/p + 1/q + 1/r will always be less than 1. The possible values for the sum (p + q + r) are 11, 10, and 9. This gives a total of three distinct possible values. Therefore, the correct option is C.