Proving Relatively Prime Property: Integers, gcd, and the Prime Divisor Problem

[scottstapp]

Homework Statement

I have to prove the following:
Let a₁,a₂, ...,a_n be integers and set b=a₁*a₂*...*a_n. If c is a nonzero integer and c is relatively prime to each a_k, then c and b are relatively prime.

Homework Equations

Definition of relatively prime: Let a and b be integers, not both zero. if gcd(a,b)=1 then a and b are relatively prime.

The Attempt at a Solution

1. Let a₁,a₂, ...,a_n be integers and let b=a₁*a₂*...*a_n. Suppose there exists a nonzero integer c where c is relatively prime to each a_k.

2. By defn, gcd(c,a_k)=1

I need help showing that the gcd(c,a₁*a₂)=1

Thanks

 

[JSuarez]

You don't need to suppose the existence of c; it is given by hyphotesis.

The best way to do this is to use induction on n and the following result (sometimes called Euclides' lemma): if c|ab and gcd(c,a)=1, then c|b.

If you haven't covered this lemma, argue like this: if gcd(c,a1*a2)>1, then there must be a common prime divisor p > 1 of c and a1*a2; p cannot divide a1 (because then gcd(a1,c)>1), so it must divide a2 and gcd(c,a2) > 1, and this is also impossible.

 

[scottstapp]

I was just looking at Euclids Lemma actually and wondering how I would prove that c|ab? Is it because c is relatively prime with each a therefore it must divide ab?

Thank you for your help once again