State/Being relatively prime to a product is equivalent to being relatively prime to the factors

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Proposition
Proof

Proposition: (Being relatively prime to a product is equivalent to being relatively prime to the factors) Suppose that $x \in \ZZ$, $x \neq 0$, and $a,b \in \ZZ$. Then the following two statements are equivalent:

$GCD(x,ab) = 1$.
$GCD(x,a) = 1$ and $GCD(x,b) = 1$.

Logical Connections

This statement logically relies on the following definitions and statements: State/Integers are relatively prime iff they have no common prime factors, State/Primes dividing a product must divide a factor

The following statements and definitions rely on the material of this page: State/Multiplicativity of the totient

To visualize the logical connections between this statements and other items of mathematical knowledge, you can visit the following cluster(s), and click the "Visualize" tab: Clust/Basic number theory

State/Being relatively prime to a product is equivalent to being relatively prime to the factors

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	Suppose that $x \in \ZZ$, $x \neq 0$, and $a,b \in \ZZ$.
	Then $GCD(x, ab) = 1$ if and only if: There does not exist a prime number $p$ such that $p$ divides $x$ and $p$ divides $ab$. Since a prime divides a product if and only if it divides a factor, we find that $GCD(x, ab) = 1$ if and only if: There does not exist a prime number $p$ such that $p$ divides $x$ and ($p$ divides $a$ or $p$ divides $b$). Therefore, $GCD(x,ab) = 1$ if and only if: There does not exist a prime number $p$ such that ($p$ divides $x$ and $p$ divides $a$) or ($p$ divides $x$ and $p$ divides $b$). Therefore, $GCD(x,ab) = 1$ if and only if both of the following are true: There does not exist a prime number $p$ such that ($p$ divides $x$ and $p$ divides $a$). There does not exist a prime number $p$ such that ($p$ divides $x$ and $p$ divides $b$). Therefore $GCD(x,ab) = 1$ if and only if both of the following are true: $GCD(x,a) = 1$. $GCD(x,b) = 1$.
	Hence $GCD(x,ab) = 1$ if and only if $GCD(x,a) = 1$ and $GCD(x,b) = 1$.