Lõpparvud

Lõpparvud (ending digits) are the digits that appear at the end of a whole number when it is written in a given numeral system, most commonly decimal. For a number N and a positive integer k, the last k digits are the residue of N modulo 10^k. Equivalently, N ≡ r (mod 10^k) where 0 ≤ r < 10^k, and r represents the last k digits, often written with leading zeros if necessary. For example, 12345 has last digit 5, last two digits 45, and last three digits 345; the number 1007 has last four digits 1007 (or 0007 if using a fixed width).

Properties and methods: The ending digits reveal information about the number’s congruence properties modulo powers of 10. They can be obtained by computing N mod 10^k or by reading the decimal representation directly. In computations with large numbers, ending digits are often preserved while truncating other digits, when only results modulo 10^k are needed.

Applications: Lõpparvud are used in divisibility tests, modular arithmetic, and the analysis of numerical sequences and patterns in endings of powers or other functions. They appear in computer algorithms that require only the least significant digits, in digital forensics and data integrity checks, and in recreational mathematics where cyclic patterns of ending digits are explored.

Generalization: The concept extends to any base b; the last k digits in base b correspond to the remainder modulo b^k. Thus, ending digits are a practical representation of modular information across bases.