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1. The remainder when (12¹³ + 23¹³) is divided by 11.

A. 0

B. 1

C. 2

D. 3

Solution:

\begin{aligned} \frac{12^{13}}{11} gives Remainder 1 \\ It can be written as \\ \frac{(12 \times 12 \times 12 \times 12 . . . . . . . . 13 times)}{11} \\ On dividing it gives remainder 1, each time . \\ 1 \times 1 \times 1 \times 1 . . . . . . . . . 13 times \\ So, final remainder will be 1 \\ \frac{23^{13}}{11} \Rightarrow Remainder 1 \\ Thus, \\ The remainder of \frac{(12^{13} + 23^{13})}{11} \\ = (1 + 1) \\ = 2 \end{aligned}

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