Division Algorithm:
The theorem stating that for any integer and positive integer , there exist unique integers (quotient) and (remainder) such that:
$
a = dq + r \quad \text{with} \quad 0 \leq r < d
$
This guarantees the existence and uniqueness of the quotient and remainder when dividing integers.
Quotients and Remainders:
The results of division, where the quotient is the integer part of the division, and the remainder is what is left over, satisfying .
Integer Division Definitions:
In the context of the Division Algorithm, the quotient and remainder are defined such that:
$
a = dq + r
$
with the specified bounds on . The quotient and remainder are uniquely determined by this relation.
Procedural Version of the Division Algorithm:
A step-by-step method to compute and for given integers and positive integer . It involves iterative subtraction or division steps to find the unique and satisfying the relation, ensuring .
1. What is the primary role of the division algorithm in number theory?
2. Who is credited with formulating the theorem that if a number divides two integers, then it divides any linear combination of those integers?
3. How do the quotient and remainder of a division fundamentally differ from each other?
Division Algorithm — statement?
Unique $q, r$ with $a = dq + r$, $0 \\leq r < d$.
Divisibility — relation?
Exists $k$ with $b = ak$.
Linear combination — form?
$ax + by$, with integers $x, y$.
Quotients and Remainders — result?
From division: $a = bq + r$, with $0 \\leq r < b$.
Modular arithmetic — relation?
$a \\equiv b \\ ( ext{mod } m)$ if $m$ divides $a - b$.
Prime number — definition?
Divisible only by 1 and itself.
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