Product of Coprime Factors
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Theorem
Let $a, b, c \in \Z$ such that $a$ and $b$ are coprime.
Let both $a$ and $b$ be divisors of $c$.
Then $a b$ is also a divisor of $c$.
That is:
- $a \perp b \land a \divides c \land b \divides c \implies a b \divides c$
Proof
By definition of divisor:
- $a \divides c \implies \exists r \in \Z: c = a r$
- $b \divides c \implies \exists s \in \Z: c = b s$
So:
\(\ds a\) | \(\perp\) | \(\ds b\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists m, n \in \Z: \, \) | \(\ds m a + n b\) | \(=\) | \(\ds 1\) | Integer Combination of Coprime Integers | |||||||||
\(\ds \leadsto \ \ \) | \(\ds c m a + c n b\) | \(=\) | \(\ds c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds b s m a + a r n b\) | \(=\) | \(\ds c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds a b \paren {s m + r n}\) | \(=\) | \(\ds c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds a b\) | \(\divides\) | \(\ds c\) | Definition of Divisor of Integer |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Some Properties of $\Z$: Exercise $2.6$
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Theorem $2 \text{-} 4$: Corollary $2$