9 को 99 बार लिख कर कोई संख्या N बनाई जाती है । यदि N को 13 से विभाजित किया जाए, तो शेषफल क्या होगा ?
Read this question in English
A number N is formed by writing 9 for 99 times. What is the remainder if N is divided by 13?
(a) 11 · (b) 9 · (c) 7 · (d) 1
उत्तर और व्याख्या · explanation in English
Answer: (a) 11
999999 (six 9s) = 999 × 1001 is divisible by 13, so each block of six 9s leaves nothing over. 99 nines are 16 such blocks followed by 999, so N leaves the same remainder as 999, which is 11.
- 999999 = 999 × 1001 = 999 × 7 × 11 × 13, so it is divisible by 13.
- 99 = 6 × 16 + 3, so N is sixteen blocks of 999999 followed by 999.
- The first 96 nines form a multiple of 999999 (and so of 13), shifted three places; hence N leaves the same remainder as 999.
- 999 = 13 × 76 + 11, so the remainder is 11.
Remember · Six identical digits (aaaaaa) are always divisible by 7, 11 and 13; remove blocks of six and work with the digits left over.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·