What is the remainder if 2¹⁹² is divided by 6?

To find the remainder when 2^192 is divided by 6, let's look at the pattern of powers of 2 modulo 6:

2^1 = 2 2^2 = 4 2^3 = 8, which is 2 (mod 6) 2^4 = 16, which is 4 (mod 6) 2^5 = 32, which is 2 (mod 6)

We observe a pattern:

• For n = 1, 2^n mod 6 = 2.
• For even n >= 2, 2^n mod 6 = 4.
• For odd n >= 3, 2^n mod 6 = 2.

The exponent in our question is 192, which is an even number and 192 >= 2. According to the pattern, when the exponent is an even number greater than or equal to 2, the remainder is 4.

Therefore, 2^192 divided by 6 leaves a remainder of 4.

Analysis of options: A) 0: Incorrect. 2^192 is a power of 2, so it only has prime factors of 2. For it to be divisible by 6, it would need a prime factor of 3, which it does not have. B) 1: Incorrect. Any power of 2 (for n>=1) is an even number. An even number cannot leave a remainder of 1 when divided by an even number like 6. C) 2: Incorrect. This would be the remainder if the exponent were an odd number greater than or equal to 3 (e.g., 2^3 = 8, 8 mod 6 = 2). However, 192 is an even exponent. D) 4: Correct. As per the pattern observed, for an even exponent n >= 2, 2^n mod 6 is 4.

The final answer is D.