Find the remainder when 54¹²⁴ is divided by 17.
a) 8
b) 13
c) 16
d) 9

Answer: b
Explanation:
$$\eqalign{ & \frac{{{{54}^{124}}}}{{17}} \cr & \frac{{{{\left( {17 \times 3 + 3} \right)}^{124}}}}{{17}} \cr & {\text{The}}\,{\text{remainder}}\,{\text{would}}\,{\text{be}}\,{\text{same}}\,{\text{as}}, \cr & \frac{{{3^{124}}}}{{17}} \cr & \frac{{{3^{4 \times 31}}}}{{17}} \cr & {\text{Remainder}}\,{\text{will}}\,{\text{be}}\,{\text{same}}\,{\text{as}}, \cr & \frac{{{3^4}}}{{17}} \cr & \frac{{81}}{{17}} \cr & \text{Remainder} = 13 \cr} $$