421 and 427, when divided by the same number, leave the same remainder 1. How many numbers can be used as the divisor in order to get the same remainder 1?
Answer & explanation
Answer: (c) 3
If both numbers leave remainder 1, the divisor divides 420 and 426, and hence their HCF, 6. The divisors of 6 that are greater than 1 are 2, 3 and 6, so three numbers work (1 cannot leave a remainder of 1).
- Remainder 1 means the divisor divides 421 − 1 = 420 and 427 − 1 = 426.
- HCF(420, 426) = HCF(420, 6) = 6.
- Divisors of 6 are 1, 2, 3 and 6. A divisor must be larger than the remainder, so 1 is ruled out.
- Possible divisors: 2, 3 and 6, i.e. 3 numbers.
- Check: 421 = 6 × 70 + 1 and 427 = 6 × 71 + 1.
Remember · Same remainder r: the divisor divides each number minus r. Count the HCF’s divisors that are larger than r.
Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·