What is the remainder when 4⁶¹ is divided by 51 ?

Solution (By Examveda Team)

4⁶¹ = 4 × 4⁶⁰ = 4 × (4⁴)¹⁵ = 4 × (256)¹⁵
Now, (xⁿ - aⁿ) is divisible by (x - a) for all values of n.
∴ (256¹⁵ - 1) is divisible by (256 - 1) i.e., 255 and hence by 51
⇒ On dividing (256)¹⁵ by 51, we get 1 as remainder
⇒ On dividing 4⁶⁰ by 51, we get 1 as remainder
⇒ On dividing 4⁶¹ by 51, remainder obtained = (4 × 1) = 4