system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: When you cube integers 1-10, the ones digit is unique for each integer (and 11-20 and 21-30...). Is there a mathematical explanation?

Fun with numbers:

Int | Cubed
-:|-:
1 | **1**
2 | **8**
3 | 2**7**
4 | 6**4**
5 | 12**5**
6 | 21**6**
7 | 34**3**
8 | 51**2**
9 | 72**9**
10 | 100**0**


In the ones digit that I bolded, you have 

    1, 8, 7, 4, 5, 6, 3, 2, 9, 0 

No repeats.

Coincidence? Mathematical explanation? Evidence of intelligent alien gods transplanting a code into the mathematical language of our universe (please be that one, please be that one)?
gpt
Any integer can be written as follows  

     x=(10a+b) s.t. 0<=b<=9

"b" is the one's digit, and "a" is all the others. (So if x=4, then a=0 and b=4, and if x=1842, then a=184 and b=2)

So,   

      x^3 =(10a+b)^3 =1000a^3 + 200^2 a^2 b+10ab^2 +100a^2 b + 20 ab^2 +b^3 = 10(100a^3 + 20^2 a^2 b+1ab^2 +10a^2 b + 2 ab^2) +b^3

More importantly x^3 = 10*[some integer]+b^3. That is, the ones digit of x^3 is the same as the ones digit of the cube of the ones digit of x (i.e. the ones digit of b^3). 

You've verified that a particular property holds for 1-10. The above proof generalizes it for all numbers.

The cube of an integer ending in 1 will always end in 1  
The cube of an integer ending in 2 will always end in 8   
The cube of an integer ending in 3 will always end in 7   
etc...