lastdigit

Lastdigit is a term used to refer to the units digit of a number in base 10, i.e., the digit in the ones place. It is equivalent to the remainder after division by 10, often written as lastdigit(n) = n mod 10. For nonnegative integers, this gives the digit from 0 to 9 that ends the decimal representation. For negative integers, the last digit is typically understood as the last digit of the absolute value, or it can be defined via the nonnegative remainder mod 10; conventions vary.

The concept generalizes to the last k digits, which are the remainder when divided by 10^k. This provides the trailing block of k digits of the number’s decimal representation and is often used in problems where only the ending digits matter.

In arithmetic, the lastdigit is compatible with modulo operations: the last digit of a sum is the last digit of the sum of the last digits, and the last digit of a product is determined by the last digits of the factors. The last digit of powers, for example, often cycles with a small period depending on the base. For instance, the last digit of powers of 7 cycles through 7, 9, 3, 1 with period 4.

Applications of the lastdigit include problem solving in number theory, pattern recognition in modular arithmetic, and programming tasks that require extracting the trailing digit(s) of a number. In programming languages, the operation is typically implemented via modulo 10 (and modulo 10^k for trailing digits). See also modulo operation and modular arithmetic.