The remainder when 4⁰ + 4¹ + 4² + 4³ + ........ + 4⁴⁰ is divided by 17 is:

Solution (By Examveda Team)

Let S be the sum of the expression, Then
S = 4⁰ + 4¹ + 4² + 4³ + ........ + 4⁴⁰
S = (1 + 4 + 16 + 64) + 4⁴ (1 = 4 + 16 + 64) + ...... + 4³⁶ + 4⁴⁰
Since, (1 + 4 + 16 + 64) = 85, is divisible by 17. Hence, except 4⁴⁰ remaining expression is divisible by 17.
Thus,
$$\frac{{{4^{40}}}}{{17}} \to \frac{{{{\left( {{4^4}} \right)}^{10}}}}{{17}} \to \frac{{{1^{10}}}}{{17}}$$
Hence, remainder = 1