If N² = 12345678987654321, then how many digits does the number N have ?
- (a) 8
- (b) 9 ✓ UPSC's answer
- (c) 10
- (d) 11
Why the answer is (b)
• 12345678987654321 has 17 digits, so 10¹⁶ ≤ N² < 10¹⁷.
• Taking square roots, 10⁸ ≤ N < √10 × 10⁸ ≈ 3.16 × 10⁸.
• Every number from 100,000,000 up to about 316,000,000 has 9 digits.
• In fact 111111111² = 12345678987654321, so N = 111111111, which has 9 digits — option (b).
Why the other options are wrong
- (a) 8
- An 8-digit number squared would result in a number with at most 16 digits, but the given number has 17 digits.
- (c) 10
- A 10-digit number squared would result in a number with at least 19 digits, which exceeds the 17 digits of the given number.
- (d) 11
- An 11-digit number squared would result in a number with at least 21 digits, which is far larger than the given 17-digit number.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.