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Cartes-fiches
Def: (divides)
If \(a,b \in \mathbb{Z}\), then a divides b, denoted by : a|b if there is an integer c with b = ac. We also say that a is a divisor of b or that b is a mulitple of a
Theorem: (Euclidean Division)
If a and b are integers with a nonzero, then there are unique integers q and r, called the quotient and remainder, with:
\(b = qa + r, \quad 0 \leq r < |a|\)
Def: (greatest common divisor)
A commen divisor of integers a and b is an integer c with:
c | a
c | b
the greatest common divisor of a and b denoted by gcd(a,b), is defined by the greatest common divisor.
Def: (Prime)
An integer p is prime if \(2 \leq p\) and its only divisors are +1,-1,-p,+p. If an integer is not prime, then it is called composite
Def: (Prime Factorization)
If \(2 \leq a\) then a prime factorization of a is
\(a = p_1^{e_1} \cdot \dots \cdot p_t^{e_t}\)
where \(p_i\) are disting primes and \(e_i \geq 0, \forall i\)
Theorem: (Euclid's Lemma)
if p is aprime and p | ab for integers a and b then
p | a or p | b
Def: (coprime)
Two integers a and b are coprime if their gcd is 1
Theorem: (Fundamental Theorem of Arithmetic)
Every integer \(a \geq 2\) has UNIQUE factorization
\(a = p_1^{e_1} \cdot \dots \cdot p_t^{e_t}\)
where \(p_1 < \dots < p_t\) and all \(p_i\) are prime and \(e_i > 0\)
