Thursday, 26 September 2019

elementary number theory - if pmida and pmidb then pmidgcd(a,b)


I would like to prove the following property :




(p,a,b)Z3pa and pbpgcd




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?

No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...