We need triples (p, q, r) of natural numbers with $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1$.

All solutions (up to permutation) are:

• (4, 4, 2): $\frac{1}{4} + \frac{1}{4} + \frac{1}{2} = 1$ → $p+q+r = 10$
• (2, 3, 6): $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1$ → $p+q+r = 11$
• (3, 3, 3): $\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1$ → $p+q+r = 9$

Distinct values of (p + q + r): {9, 10, 11} → 3 values.

Hence, the answer is Three.