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Question

For all natural numbers n, 23n7n1 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 23n7n1, we get
2371=0divisible by all positive integers

Substitute n=2 in 23n7n1, we get
26141=49
Let P(n):23n7n1 is divisible by 49
P(2) is true.
Assume P(k) is true
23k7k1=49m
Substituting k+1 in place of n, we get
23k+37(k+1)1=8.23k7k8=8.(23k7k1)+7.7k=49(8m+k)divisible by 49
P(k+1) is true

Hence, P(n) is true.


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