I would like to prove the following property :
∀(p,a,b)∈Z3p∣a and p∣b⟹p∣gcd(a,b)
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 Div(a) denotes the set of divisors of a, the greatest common divisor of a and b is gcd(a,b)=max(Div(a)∩Div(b))
- d=gcd(a,b)⟺{d∈Div(a)∩Div(b)∀x∈Div(a)∩Div(b):x≤d
- ∀(a,b)∈N2a∣b⟺Div(a)⊂Div(b)
- ∀x∈ZDiv(x)=Div(−x)
- If a,b∈Z, then gcd(a,b)=gcd(|a|,|b|), adding gcd(0,0)=0
Indeed,
Let (p,a,b)∈Z3 such that p∣a and p∣b then :
p∣a⟺Div(p)⊂Div(a) and p∣b⟺Div(p)⊂Div(b) then
Div(p)⊂(Div(a)∩Div(b))⟺p∣gcd(a,b)
Am I right?
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