First slide

Binomial theorem for positive integral Index

Question

The remainder when $6^{n} - 5^{n}$ is divided by $25$ , is

Moderate

A

1
B

24
C

0
D

$n$

Solution

$\begin{array}{l} 6^{n} - 5 n = (1 + 5)^{n} - 5 n \\ = 1 + 5 n +^{n} C_{2} (5^{2}) +^{n} C_{3} (5^{3}) + \dots +^{n} C_{n} (5^{n}) \\ - 5 n \\ = 1 + 25 m \end{array}$
Where $m =^{n} C_{2} +^{n} C_{3} (5) + \dots +^{n} C_{n} (5^{n - 2})$ is a whole number.
Thus, remainder when $6^{n} - 5^{n}$ is divided by $25$ is $1 .$

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