I would like to prove the following property :
∀(p,a,b)∈Z3p∣a and p∣b⟹p∣gcd
Knowing that :
Definition
Given two natural numbers a and b, not both zero, their greatest common divisor is the largest divisor of a and b.
- If \operatorname{Div}(a) denotes the set of divisors of a, the greatest common divisor of a and b is \gcd(a,b)=\max(\operatorname{Div}(a)\cap\operatorname{Div}(b))
- d=\operatorname{gcd}(a,b)\iff \begin{cases}d\in \operatorname{Div}(a)\cap\operatorname{Div}(b) & \\ & \\ \forall x \in \operatorname{Div}(a)\cap\operatorname{Div}(b): x\leq d \end{cases}
- \forall (a,b) \in \mathbb{N}^{2}\quad a\mid b \iff Div(a) \subset Div(b)
- \forall x\in \mathbb{Z}\quad \operatorname{Div}(x)=\operatorname{Div}(-x)
- If a,b\in\mathbb{Z}, then \gcd(a,b)=\gcd(|a|,|b|), adding \gcd(0,0)=0
Indeed,
Let (p,a,b)\in\mathbb{Z}^{3} such that p\mid a and p\mid b then :
p\mid a \iff \operatorname{Div}(p)\subset \operatorname{Div}(a) and p\mid b \iff \operatorname{Div}(p)\subset \operatorname{Div}(b) then
\operatorname{Div}(p)\subset \left( \operatorname{Div}(a)\cap \operatorname{Div}(b)\right) \iff p\mid \gcd(a,b)
Am I right?
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