## Coprimality with a product of integers

## The theorem

If and are such that is separately coprime with each , then is coprime with their product .

## The proof

Proof: Through Bézout. being coprime with each , we can write, for each :

If we take the product of each of these expressions, we get:

Among the terms of this product, all but one contain as a factor. The one that doesn’t have as a factor is . Hence we obtain an expression of the form:

Hence and are coprime.

## In the case of polynomials

The same result holds with polynomials: if and are polynomials over a field , such that is coprime with each , then is coprime with their product .

The proof is identical.