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Finding Remainder of Numbers of the Form a^b

The statement...

Question

The statement $P (n) = 9^{n} - 8^{n}$ when divided $8$ always leaves the remainder

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Solution

The correct option is B $1$
$P (n) = 9^{n} - 8^{n} = {(1 + 8)}^{n} - 8^{n} = 1 + n_{C_{1}} 8 + n_{C_{2}} 8^{2} + . . . + n_{C_{n}} 8^{n} - 8^{n}$

$\Rightarrow P (n) = 1 + 8 (n_{C_{1}} + n_{C_{2}} 8 + . . . + n_{C_{n}} 8^{n - 1} - 8^{n - 1})$

Therefore, when divided by $8$ , the remainder we get is $1$
Ans: C

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