Prove by induction 3|5^(2n)-1

Question

I have been trying to prove this question using induction, but the furthest I get is solving the base case

Answer 1

Prove that 3 is a divisor of 5²ⁿ - 1 for all positive integers n.

 

   Step 1:  show that the statement is true for n = 1.

 

                If n = 1, then 5²ⁿ - 1 = 5² - 1 = 24, which is divisible by 3.

 

   Step 2:  Assume that 3 is a divisor of 5²ⁿ -1.  Show that 3 is a divisor of 5²⁽ⁿ⁺¹⁾ - 1

 

                5²⁽ⁿ⁺¹⁾ - 1 = 5²ⁿ5² - 1

 

                                 = 25(5²ⁿ) - 1 = (3·8 + 1)(5²ⁿ) - 1

 

                                                      = 3·8(5²ⁿ) + 5²ⁿ - 1

 

                                                      = 3·8(5²ⁿ) + 3k, for some integer k (by induction hypothesis)

 

                                                      = 3[8(5²ⁿ) + k)]  (Since 8(5²ⁿ)+k is an integer, 5²⁽ⁿ⁺¹⁾ - 1 is divisible by 3).