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 neces…

Question

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)?

1. A. None
2. B. One
3. C. Three (Correct answer)
4. D. More than three

Correct Answer

Option C — Three

Detailed Solution & Explanation

The correct answer is Three.

Key Points

• 1/p + 1/q + 1/r = 1 with natural numbers.
• WLOG p ≤ q ≤ r. Then 1/p ≥ 1/3, so p ≤ 3. If p=2: 1/q+1/r=1/2 → (q,r) = (3,6),(4,4).
• If p=3: 1/q+1/r=2/3 → q=3,r=3 works (1/3+1/3+1/3=1).
• Solutions: (2,3,6),(2,4,4),(3,3,3) — exactly three. Answer: Three.