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?
Answer & explanation
Answer: (c) 8
Dividing N = p + q + r by N turns the condition into 1 = 1/a + 1/b + 1/c with three different whole numbers, and the only such solution is 1/2 + 1/3 + 1/6. So N must be a multiple of 6, and there are 8 multiples of 6 below 50.
- Divide N = p + q + r by N: 1 = p/N + q/N + r/N = 1/a + 1/b + 1/c, where a = N/p, b = N/q, c = N/r are whole numbers.
- Distinct factors mean a, b, c are distinct. Take a < b < c. If a ≥ 3, the sum is at most 1/3 + 1/4 + 1/5 < 1, so a = 2.
- Then 1/b + 1/c = 1/2 with 2 < b < c: b = 3 gives c = 6; b = 4 gives c = 4 (not distinct); b ≥ 5 is too small. So (a, b, c) = (2, 3, 6).
- Hence N must be divisible by 2, 3 and 6, i.e. a multiple of 6, and then N = N/2 + N/3 + N/6 always works.
- Multiples of 6 below 50: 6, 12, 18, 24, 30, 36, 42, 48 — that is 8 numbers.
- Check: 6 = 3 + 2 + 1 and 48 = 24 + 16 + 8.
Remember · When a number equals a sum of its own factors, divide through by it — the problem becomes unit fractions adding to 1.
Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·