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Sum of binomial coefficients with alternate signs

9n +7 is divi...

Question

$9^{n}$ + 7 is divisible by ________.

A

4

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B

8

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C

3

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D

6

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Solution

The correct options are
A
4

B
8

$9^{n}$ + 7 = $(1 + 8)^{n} + 7$

$(1 + 8)^{n} + 7$ = $(^{n} C_{0} (8)^{n} +^{n} C_{1} (8)^{n - 1} +^{n} C_{2} (8)^{n - 2} . . . . . . . . . . . . +^{n} C_{n - 1} (8)^{1} +^{n} C_{n}) + 7$

$(1 + 8)^{n} + 7$ = $(^{n} C_{0} (8)^{n} +^{n} C_{1} (8)^{n - 1} +^{n} C_{2} (8)^{n - 2} . . . . . . . . . . . . +^{n} C_{n - 1} (8)^{1} + 1) + 7$

$(1 + 8)^{n} + 7$ = $(^{n} C_{0} (8)^{n} +^{n} C_{1} (8)^{n - 1} +^{n} C_{2} (8)^{n - 2} . . . . . . . . . . . . +^{n} C_{n - 1} (8)^{1} + 8$

= 8k

$\Rightarrow$ It is divisible by 8 and 4

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