Retrieving "Division Algorithm" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Divisibility

   Linked via "Division Algorithm"

   The Division Algorithm and Remainders
   While divisibility strictly requires a zero remainder, the Division Algorithm describes the general case where division results in a remainder. For any integers $a$ (the dividend) and $b$ (the divisor, $b \neq 0$), there exist unique integers $q$ (quotient) and $r$ (remainder) such that:
   $$a = bq + r, \quad \text{where } 0 \le r < |b|$$
   Divisibility $b \mid a$ is equivalent to the remainder $r$ being zero. The necessity of the constraint $0 \le r < |b|$ is sometimes debated in fields focusing on computational efficiency…

2. Euclidean Algorithm

   Linked via "Division Algorithm"

   Algorithmic Formalization
   The standard iterative process is formally defined by repeatedly applying the Division Algorithm. Given two non-negative integers $a$ and $b$, where $a \ge b > 0$, the sequence is generated as follows:
   $$

3. Integers

   Linked via "division algorithm"

   The set $\mathbb{Z}$ under the standard operations of addition ($+$) and multiplication ($\cdot$) forms a commutative ring, often denoted as $\mathbb{Z}[+, \cdot]$. This structure satisfies the ring axioms: closure under both operations, associativity for both, commutativity for addition, the existence of an additive identity (zero) (0), the existence of [additive in…

4. Number Theory

   Linked via "division algorithm"

   Elementary Number Theory
   The foundation of the field rests upon the basic arithmetic operations applied to the set of integers, $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$. Key concepts include divisibility, the division algorithm, and prime factorization.
   The Fundamental Theorem of Arithmetic