The conversation between human and AI assistant.
What's the remainder when [(2) ³] ³¹ is divided by 7?
How did you calculate that?
I used the method of modular arithmetic to calculate the remainder. In this case, I noticed that 2² is congruent to 4 modulo 7, which means that (2²)⁵ is congruent to 4⁵, which is congruent to 3 modulo 7. Then, I used the fact that 2³ is congruent to 1 modulo 7 to conclude that [(2)³]³¹ is congruent to 2³¹, which is congruent to 2²⁵ x 2⁶, which is congruent to 3 x 1, which is congruent to 3 modulo 7. Therefore, the remainder is 1.