Mod10

Mod10, or arithmetic modulo 10, refers to the arithmetic of integers under the equivalence relation a ≡ b (mod 10) when their difference is a multiple of 10. The residue classes are [0], [1], ..., [9], often represented by the last digit of an integer. Two integers are congruent modulo 10 if they share the same last decimal digit. The set Z/10Z forms a ring: addition and multiplication are performed modulo 10, and the result is reduced to the range 0 through 9. Ten is composite, so the ring has zero divisors (for example, 2·5 ≡ 0 mod 10) and is not a field. An element has a multiplicative inverse modulo 10 only if it is coprime to 10; the units are 1, 3, 7, and 9, with inverses 1, 7, 3, and 9 respectively.

In practice, modulo 10 arithmetic is often used to track the last digits in decimal computations. It also underpins certain checksum schemes like the final-digit constraint found in some error-detection algorithms. Equations of the form a ≡ b (mod 10) can be solved by replacing a with b plus or minus multiples of 10; equivalently, a and b have the same remainder upon division by 10. The Chinese remainder theorem shows Z/10Z is isomorphic to Z/2Z × Z/5Z, reflecting its decomposition into simpler components.