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

Question 629881: prove that 4 is a factor of 5^n-1 for all n
Answer by reviewermath(1029) (Show Source): You can put this solution on YOUR website!
Proof by Mathematical Induction.
Check if true for n = 1
, true because 4 is a factor of 4.
Assume that P(k) is true, that is
4 is a factor of .
We need to show that P(k+1) is true. That 4 is a factor of .
Since 4 is a factor of , then there is some positive integer q such that .

=
= , substitute
=
=
Let p = 5q + 1, p is a positive integer because positive integers are closed under addition and multiplication.
Since for some positive integer p, then 4 is a factor of
. Therefore, P(k+1) is true and we have proven that 4 is a factor of for all positive integer n.
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