A number N is formed by writing 9 for 99 times. What is the remainder if N is divided by 13 ?
- (a) 11 ✓ UPSC's answer
- (b) 9
- (c) 7
- (d) 1
Why the answer is (a)
• N (ninety-nine 9s) = 10⁹⁹ − 1.
• Remainders of powers of 10 on division by 13: 10 → 10, 10² → 9, 10³ → 12 (that is, −1), so 10⁶ leaves 1 and the pattern repeats every 6.
• 99 = 6 × 16 + 3, so 10⁹⁹ leaves the same remainder as 10³, i.e. 12.
• N = 10⁹⁹ − 1 therefore leaves 12 − 1 = 11, option (a).
Why the other options are wrong
- (b) 9
- 10⁹⁹ − 1 leaves remainder 11 on division by 13, not 9.
- (c) 7
- 10⁹⁹ − 1 leaves remainder 11 on division by 13, not 7.
- (d) 1
- 10⁹⁹ − 1 leaves remainder 11 on division by 13, not 1.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.