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

Answer: Option E

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