What will be the unit digit in the 7¹⁰⁵ ?

Solution (By Examveda Team)

$$\eqalign{ & {{\text{7}}^{105}} \Rightarrow \cr & {{\text{7}}^1} \to 7 \to 7 \cr & {{\text{7}}^2} \to 49 \to 9 \cr & {{\text{7}}^3} \to 343 \to 3 \cr & {{\text{7}}^4} \to 2401 \to 1 \cr & {\text{Divided power of 7 by 4}} \cr & \frac{{105}}{4} \to {\text{ remainder = 1}} \cr & \Rightarrow {{\text{7}}^1}\,{\text{is left unit digit = 1}} \times 7 = 7 \cr} $$