How can numbers be added in arithmetic modulo 10?

In modular arithmetic, numbers are added by performing the standard arithmetic sum and then reducing the result to a value between 0 and $N-1$, where $N$ is the modulus.

For arithmetic modulo 10, you follow these steps:

• Calculate the sum: Add the numbers together using standard addition.
• Reduce by the modulus: Subtract the modulus (10) from the sum as many times as necessary until the result is a number $M$ such that $0\le M<10$. This final value $M$ is the sum modulo 10.

For example, using the notation introduced by Carl Friedrich Gauss, the sum $2+4+3+7$ equals 16. Since $16-10=6$, we write:
$$2+4+3+7\equiv 6\left(\mathrm{mod}10\right)$$

This process is a fundamental aspect of number theory, where modular arithmetic serves as a tool for solving Diophantine equations and understanding congruence classes.1