First slide

Binomial theorem for positive integral Index

Question

When 5⁹⁹ is divided by 13, the remainder is

Moderate

A

8
B

9
C

10
D

none of these

Solution

We have,
$\begin{aligned} 5^{99} = 5^{3} \cdot 5^{96} = (125) (625)^{24} \\ = [13 \times 9 + 8] (1 + 48 \times 13)^{24} \\ = (13 \times 9 + 8) [1 +^{24} C_{1} \times (48 \times 13) +^{24} C_{2} (48 \times 13)^{2} + \dots + (48 \times 13)^{24}] \end{aligned}$
= 8 + terms containing powers of 13. Hence remainder = 8.

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