Find the remainder when $(1 + 8)^{n} + 7$ is divided by 8

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Solution

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

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

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

= 8k

So the remainder will be 0

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