Which of the following numbers will completely divide $(6^{61}+6^{62}+6^{63}+6^{64})$?

Let's factor out the smallest power of $6$ from the given expression:
$$E=6^{61}+6^{62}+6^{63}+6^{64}$$
$$E=6^{61}\cdot (1+6^1+6^2+6^3)$$
Now, simplify the terms inside the parentheses:
$$1+6+36+216=259$$
So, the expression becomes:
$$E=6^{61}\times 259$$
Let's test the options to see which number completely divides $259$ or a combination with a factor of $6$:Checking option A ($10$): $E=6^{60}\times 6\times 259=6^{60}\times 1554$, which ends in $4$ and is not divisible by $10$.Checking option B ($13$): Let's divide $259$ by $13$: $\frac{259}{13}=19.92$ (not completely divisible).Checking option C ($7$): Let's divide $259$ by $7$: $\frac{259}{7}=37$ (completely divisible!).Since $259$ is perfectly divisible by $7$, the entire expression is completely divisible by $7$.