Question

DO NOT GIVE WRONG ANSWER.

Answer 1

$4^{61} + 4^{62} + 4^{63} + 4^{64} + 4^{65} \\ \\ = 4^{61 + 0} + 4^{61 + 1} + 4^{61 + 2} + 4^{61 + 3} + 4^{61 + 4} \\ \\ = (4^{61} \times 4^0) + (4^{61} \times 4^1) + (4^{61} \times 4^2) + (4^{61} \times 4^3) + (4^{61} \times 4^4) \\ \\ = (4^{61} \times 1) + (4^{61} \times 4) + (4^{61} \times 16) + (4^{61} \times 64) + (4^{61} \times 256) \\ \\ = 4^{61}(1 + 4 + 16 + 64 + 256) \\ \\ = 4^{61} \times 341$

Here, we can find that the sum given in the question is the multiple of 4^61 and 341.

341 can't be divided by 3, 6 and 13. So the sum can't be divided by these numbers.

But 341 can be divided by 11.

341 ÷ 11 = 31

So the sum can be divided by 11.

When the sum is divided by 11, we get 4^61 × 31.

∴ 11 is the answer.

Option (D) is the answer.

Hope this is understood by you.

Please mark my answer as the brainliest if this may be helpful.

Thank you. Have a nice day.

#adithyasajeevan