Why is there a pattern to the last digits of square numbers? I was programming and I realized that the last digit of all the integer numbers squared end in $ 0, 1, 4, 5, 6,$ or $ 9 $.
And in addition, the numbers that end in $ 1, 4, 9, 6 $ are repeated twice as many times as the numbers that end in $ 0, 5$
I checked the numbers from $1$ to $1000$, and the results are:
$1.$ The numbers on the left are the last digit of each digit squared.
$2.$ The numbers on the right are the number of times that the last digit is repeated.
$$
\begin{array}{cc}
0: &100, \\
1: &200, \\
4: &200, \\
5: &100, \\
6: &200, \\
9: &200
\end{array}
$$
So, why does this happen? What is the property that all integers have?

What you are looking at is the residues of squares modulo $10$.
$0^2=\color{red}0\bmod 10\\1^2=\color{blue}1\bmod 10\\2^2=\color{orange}4\bmod 10\\3^2=9\bmod 10\\4^2=\color{green}6\bmod 10\\5^2=\color{brown}5\bmod 10\\6^2=\color{green}6\bmod 10\\7^2=9\bmod 10\\8^2=\color{orange}4\bmod 10\\9^2=\color{blue}1\bmod 10$
As you can see, $0$ and $5$ are half as frequent as the other residues which are indeed $1,4,6,9$.
