proof that 5^20 - 5^19 is even without using calculator

Question

Without using calculator kindly help proof the question, it involves prime power proof.

Answer 1

Simply factoring:

    5²⁰ - 5¹⁹ = 5¹⁹(5 - 1) = 5¹⁹(4), which is divisible by 2 (that is, it is even)

Answer 2

To prove it is even, it is sufficient to show it equals 2n, where n is any integer.

 

5²⁰ - 5¹⁹ = 5¹⁹(5¹ - 5⁰) = 5¹⁹(5 - 1) = 5¹⁹(4) = 2[2(5¹⁹)]

 

Let n = 2(5¹⁹)

 

Then

 

5²⁰ - 5¹⁹ = 2n, i.e. a multiple of 2, so it is even