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Question

# For all natural numbers n, 23nâˆ’7nâˆ’1 is divisible by

A

64

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B

36

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C

49

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D

25

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Solution

## The correct option is C 49 Substitute n=1 in 23n−7n−1, we get 23−7−1=0→divisible by all positive integers Substitute n=2 in 23n−7n−1, we get 26−14−1=49 Let P(n):23n−7n−1 is divisible by 49 P(2) is true. Assume P(k) is true 23k−7k−1=49m Substituting k+1 in place of n, we get 23k+3−7(k+1)−1=8.23k−7k−8=8.(23k−7k−1)+7.7k=49(8m+k)→divisible by 49 P(k+1) is true Hence, P(n) is true.

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