SOLUTION: prove that 4 is a factor of 5^n-1 for all n

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Question 629881: prove that 4 is a factor of 5^n-1 for all n
Answer by reviewermath(1029) About Me  (Show Source):
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Proof by Mathematical Induction.
Check if true for n = 1
5%5E1+-+1+=+4, true because 4 is a factor of 4.
Assume that P(k) is true, that is
4 is a factor of 5%5Ek+-+1.
We need to show that P(k+1) is true. That 4 is a factor of 5%5E%28k%2B1%29+-+1.
Since 4 is a factor of 5%5Ek+-+1, then there is some positive integer q such that 5%5Ek+-+1+=+4q.
5%5E%28k%2B1%29+-+1+=+5%285%5Ek%29+-+1
= 5%285%5Ek%29+-+5+%2B+4
= 5%285%5Ek+-+1%29+%2B+4, substitute 5%5Ek+-+1+=+4q
= 5%284q%29+%2B+4
= 4%285q+%2B+1%29
Let p = 5q + 1, p is a positive integer because positive integers are closed under addition and multiplication.
Since 5%5E%28k%2B1%29+-+1+=+4p for some positive integer p, then 4 is a factor of
5%5Ek+-+1. Therefore, P(k+1) is true and we have proven that 4 is a factor of 5%5En-1 for all positive integer n.