What is the remainder if 2¹⁹² is divided by 6?

Answer: (d) 4

Divided by 6, the powers of 2 leave remainders 2, 4, 2, 4 … in turn: odd powers leave 2 and even powers leave 4. Since 192 is even, the remainder is 4.

1. 2¹ = 2 leaves 2; 2² = 4 leaves 4; 2³ = 8 leaves 2; 2⁴ = 16 leaves 4.
2. The remainders repeat in a cycle of 2: odd powers leave 2, even powers leave 4.
3. 192 is even, so 2¹⁹² leaves remainder 4.
4. Check: 2¹⁹² = 4⁹⁶ leaves 1 on division by 3, and it is even; of 2 and 4, only 4 is even and leaves 1 on division by 3.

Remember · For remainders of large powers, list the first few remainders, spot the cycle, then place the exponent in it.

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). · Report an error