From given options the number that will completely divide $$(3^{61}+3^{62}+3^{63}+3^{64})$$​ is:

Correct option is C

Given:

$$(3^{61}+3^{62}+3^{63}+3^{64})$$

Solution:

$$(3^{61}+3^{62}+3^{63}+3^{64})$$= 3⁶¹(1 + 3 + 3² + 3³)=3⁶¹(40) = 3⁶¹ ​$$\times$$​2³​$$\times$$​ 5
Let's examine each option:

A. 11: There's no factor of 11 in our simplified expression. So, 11 will not divide the expression completely.

B. 16: 16 is ($$2^4$$​). We only have ($$2^3$$​) in our expression. Thus, 16 will not divide the expression completely.

C. 30: $$30= 2\times 3\times 5$$​. We have 2, 3, and 5 as factors in our expression 3⁶¹ ​$$\times$$​2³​$$\times$$ 5  . Therefore, 30 will divide the expression completely.​ 

D. 25: 25 is (5²). We only have 5 as a factor in our expression (not (5²)). So, 25 will not divide the expression completely.

The correct answer is C. 30