Dear All My Friends,
Why fifth power of any digit is ended with this same digit also?
Here is my calculations:
0^5 = 0
1^5 = 1
2^5 = 32
3^5 = 243
4^5 = 1024
5^5 = 3125
6^5 = 7776
7^5 =16807
8^5 = 32768
9^5 = 59049
Thank you and best regards,
Bui Quang Tuan

>Why fifth power of any digit is ended with this same digit
>also?

According to Fermat's Little Theorem

ap ≡ a (mod p)

for any a not divisible by p, p a prime.

Well, 5 is a prime and, for any a, and for any p

ap - a is even,

so that, for any odd prime,

ap ≡ a (mod 2p)

or, in case of p = 5,

ap ≡ a (mod 10)

meaning, in part, that a and ap end with the same digit.

In the exceptional of a divisible by 5, its powers have the same last digit anyway.

Thank you for interesting proof by using interesting theorem. I have found two following proofs:

1. y = x^5 - x = (x - 1)*x*(x + 1)*(x^2 + 1)
Therefore y = 0 (mod 5) and y = 0 (mod 2). It means x^5 and x are ended with the same digit.

2. (x + 1)^5 = (x^5 + 5*x^4 + 10*x^3 + 10*x^2 + 5*x + 1) =
5*x*(1 + x)*(1 + x + x^2) + (x^5 + 1)
Easy to show 5*x*(1 + x)*(1 + x + x^2) = 0 (mod 10)
Therefore the last digit of (x + 1)^5 is the last digit of x^5 plus 1. Start with x = 0 we can get all the last digits of 0, 1, 2, ... 9 are the same 0, 1, 2,...9

May be exist also another interesting proofs?

I am thinking now about two small questions:
1. Why after one, five is the first digit with this property?
2. Why other one, only 5 and 6 hold the property: the last digit of x^n is the same x with any n?

Best regards,
Bui Quang Tuan

>1. Why after one, five is the first digit with this
>property?

Because 10 = 2*5 and 2 is too small.

>2. Why other one, only 5 and 6 hold the property: the last
>digit of x^n is the same x with any n?

Base 
6
3² = 13

4² = 24
10
5² = 25

6² = 36
12
4² = 14

9² = 69
14
7² = 37

8² = 48
15
6² = 26

A² = 6A
18
9² = 49

A² = 5A
21
7² = 27