Question

$1^{13}+2^{13}+3^{13}+.....+{60}^{13}$ is divisible by:

• A. $61$
• B. $63$
• C. $65$
• D. $59$

Answer 1

The correct option is A

$61$

Stating the given part and Solving the given expression:

Concept used: $a^n+b^n$ is divisible by $(a+b)$, when $n$ is odd.

Given:

$$\begin{gathered} =1^{13}+2^{13}+3^{13}+.....+{60}^{13} \\ =(1^{13}+{60}^{13})+(2^{13}+{59}^{13}).......+({30}^{13}+{31}^{13}) \end{gathered}$$

Here,$n=13$ which is an odd number. Hence each term is divisible by $61.$

Final answer: $1^{13}+2^{13}+3^{13}+.....+{60}^{13}$ is divisible by $\mathbf{61}$.