When a certain number is multiplied by 7, the product entirely comprises ones only (1111...). What is the smallest such number?
Answer & explanation
Answer: (d) 15873
The smallest number made only of 1s that 7 divides is 111111 (six ones). 111111 ÷ 7 = 15873.
- Divide 1, 11, 111, … by 7: the remainders are 1, 4, 6, 5, 2, 0 — the first zero comes at 111111.
- 111111 ÷ 7 = 15873.
- Check: 15873 × 7 = 111111.
- Useful fact: 111111 = 3 × 7 × 11 × 13 × 37.
Remember · Divide repunits (1, 11, 111 …) by 7 until the remainder is 0; 111111 = 7 × 15873.
Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·