What is the remainder if 2¹⁹² is divided by 6 ?
- (a) 0
- (b) 1
- (c) 2
- (d) 4 ✓ UPSC's answer
Why the answer is (d)
• Powers of 2 on division by 6 leave remainders 2, 4, 2, 4, … : 2¹ → 2, 2² → 4, 2³ = 8 → 2, 2⁴ = 16 → 4.
• Odd powers leave 2 and even powers leave 4.
• 192 is even, so 2¹⁹² leaves remainder 4.
• Hence option (d).
Why the other options are wrong
- (a) 0
- 2¹⁹² is not divisible by 3, so it cannot leave remainder 0 on division by 6.
- (b) 1
- 2¹⁹² is even, so the remainder on division by 6 cannot be odd.
- (c) 2
- Remainder 2 occurs only for odd powers of 2; 192 is even.
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.