The number 2 5^64 × 64^25 is the square of a natural number n. The sum of the digits of n is = ?

Answer: Option B

$$\begin{array}{rl}& \iff {\text{n}}^2={\left(\text{25}\right)}^{64}\times {\left(64\right)}^{25}\\ & \iff {\text{n}}^2={\left({\text{5}}^2\right)}^{64}\times {\left(2^6\right)}^{25}\\ & \iff {\text{n}}^2=5^{128}\times 2^{150}\\ & \iff {\text{n}}^2=5^{128}\times 2^{128}\times 2^{22}\\ & \iff n=5^{64}\times 2^{64}\times 2^{11}\\ & \iff n={\left(5\times 2\right)}^{64}\times 2^{11}\\ & \iff n=10^{64}\times 2048\end{array}$$
∴ Sum of digits of n
= 2 + 0 + 4 + 8
= 14