What is the remainder if $2^192$ is divided by 6?

Correct Answer: (D) 4

Examining the powers of 2 modulo 6: $2^1 = 2$$\equiv$ 2, $2^2 = 4$$\equiv$ 4, $2^3 = 8$$\equiv$ 2, $2^4 = 16$$\equiv$ 4, $2^5 = 32$$\equiv$ 2, and $2^6 = 64$$\equiv$ 4 \pmod 6. For every positive even exponent $n$, $2^n$$\equiv$ 4 \pmod 6. Because 192 is an even integer, $2^{192}$ divided by 6 leaves a remainder of 4.