What is the remainder if 2^192 is divided by 6 ?
- (a) 0
- (b) 1
- (c) 2
- (d) 4 ✓ UPSC's answer
Why the answer is (d)
• Note that 2^n is always even for n ≥ 1, so the remainder upon division by 6 must be an even number: 0, 2, or 4.
• Check small powers of 2 modulo 6: 2^1 = 2, 2^2 = 4, 2^3 = 8 ≡ 2, 2^4 = 16 ≡ 4.
• The pattern of remainders for 2^n mod 6 alternates: 2 for odd n, 4 for even n.
• Since 192 is even, 2^192 ≡ 4 (mod 6).
• Therefore, the remainder is 4, which corresponds to option (d).
Why the other options are wrong
- (a) 0
- 2^192 is not divisible by 3, so it cannot leave a remainder of 0 when divided by 6.
- (b) 1
- 2^192 is even, so it cannot leave an odd remainder like 1 when divided by 6.
- (c) 2
- The remainder 2 occurs only for odd exponents, but 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.