Integer Coprime to all Factors is Coprime to Whole

Theorem
Let $a, b \in \Z$.

Let $\ds b = \prod_{j \mathop = 1}^r b_j$

Let $a$ be coprime to each of $b_1, \ldots, b_r$.

Then $a$ is coprime to $b$.

Proof
From Integer Combination of Coprime Integers:
 * $\forall j \in \set {1, 2, \ldots, r}: a x_j + b_j y_j = 1$

for some $x_j, y_j \in \Z$.

Thus:
 * $\ds \prod_{j \mathop = 1}^r b_j y_j = \prod_{j \mathop = 1}^r \paren {1 - a x_j}$

But $\ds \prod_{j \mathop = 1}^r \paren {1 - a x_j}$ is of the form $1 - a z$.

Thus:

and so, by Integer Combination of Coprime Integers, $a$ and $b$ are coprime.

Also see

 * Integer Coprime to Factors is Coprime to Whole: the specific case where $r = 2$