Divisor Relation is Antisymmetric/Corollary

Corollary to Divisor Relation is Antisymmetric

Let $a, b \in \Z$.

If $a \divides b$ and $b \divides a$ then $a = b$ or $a = -b$.

Proof 1

Let $a \divides b$ and $b \divides a$.

Then from Divisor Relation is Antisymmetric:

$\size a = \size b$

The result follows from Integer Divides its Negative and Integer Divides its Absolute Value.

$\blacksquare$

Proof 2

Let $a \divides b$ and $b \divides a$.

Then by definition of divisor:

$\exists c, d \in \Z: a c = b, b d = a$

Thus:

\(\ds a c d\)

\(=\)

\(\ds a\)

\(\ds \leadsto \ \ \)

\(\ds c d\)

\(=\)

\(\ds 1\)

\(\ds \leadsto \ \ \)

\(\ds d\)

\(=\)

\(\ds \pm 1\)

Divisors of One

\(\ds \leadsto \ \ \)

\(\ds a\)

\(=\)

\(\ds \pm b\)

as $a = b d$

$\blacksquare$

Sources

• 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 $2 \text{-} 2 \ (5)$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$: $\mathbf D. \, 3$