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.
Prime factorization — theorem?
Unique product of primes for each integer > 1.
GCD — meaning?
Largest divisor common to two numbers.
Euclid's Algorithm — purpose?
Efficient GCD computation via division.
Extended Euclidean Algorithm — finds?
GCD and coefficients for $ax + by = \\gcd(a, b)$.
Multiplicative inverse — condition?
Exists if and only if $a$ and $m$ are coprime.
Division Algorithm — applies to?
All integers, positive divisor.
Divisibility — extended to?
Linear combinations; divisibility of sums.
Prime numbers — importance?
Building blocks of integers.
Prime factorization — uniqueness?
Yes, up to order.
GCD — computed by?
Prime factorization or Euclid's Algorithm.
Modular arithmetic — properties?
Addition, subtraction, multiplication preserve congruence.
Linear combination — purpose?
Express GCD as $ax + by$.
Inverse — exists when?
When $\\gcd(a, m) = 1$.
Key concept — in number theory?
Prime factorization and divisibility.
Teste tes connaissances avec un QCM de 10 questions sur Number Theory Fundamentals.
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?
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