Question 629881
Proof by Mathematical Induction.
Check if true for n = 1
{{{5^1 - 1 = 4}}}, true because 4 is a factor of 4.
Assume that P(k) is true, that is
4 is a factor of {{{5^k - 1}}}.
We need to show that P(k+1) is true. That 4 is a factor of {{{5^(k+1) - 1}}}.
Since 4 is a factor of {{{5^k - 1}}}, then there is some positive integer q such that {{{5^k - 1 = 4q}}}.
{{{5^(k+1) - 1 = 5(5^k) - 1}}}
               = {{{5(5^k) - 5 + 4}}}
               = {{{5(5^k - 1) + 4}}}, substitute {{{5^k - 1 = 4q}}}
               = {{{5(4q) + 4}}}
               = {{{4(5q + 1)}}}
Let p = 5q + 1, p is a positive integer because positive integers are closed under addition and multiplication.
Since {{{5^(k+1) - 1 = 4p}}} for some positive integer p, then 4 is a factor of
{{{5^k - 1}}}. Therefore, P(k+1) is true and we have proven that 4 is a factor of {{{5^n-1}}} for all positive integer n.