Divisor Relation is Transitive

Theorem

The divisibility relation is a transitive relation on $\Z$, the set of integers.

That is:

$\forall x, y, z \in \Z: x \divides y \land y \divides z \implies x \divides z$

Proof 1

We have that Integers form Integral Domain.

The result then follows directly from Divisor Relation in Integral Domain is Transitive.

$\blacksquare$

Proof 2

$\ds x$

$\divides$

$\ds y$

$\ds \leadsto \ \ $

$\ds \exists q_1 \in \Z: \, $

$\ds q_1 x$

$=$

$\ds y$

Definition of Divisor of Integer

$\ds y$

$\divides$

$\ds z$

$\ds \leadsto \ \ $

$\ds \exists q_2 \in \Z: \, $

$\ds q_2 y$

$=$

$\ds z$

Definition of Divisor of Integer

$\ds \leadsto \ \ $

$\ds q_2 \paren {q_1 x}$

$=$

$\ds z$

substituting for $y$

$\ds \leadsto \ \ $

$\ds \paren {q_2 q_1} x$

$=$

$\ds z$

Integer Multiplication is Associative

$\ds \leadsto \ \ $

$\ds \exists q \in \Z: \, $

$\ds q x$

$=$

$\ds z$

where $q = q_1 q_2$

$\ds \leadsto \ \ $

$\ds x$

$\divides$

$\ds z$

Definition of Divisor of Integer

$\blacksquare$

Sources

1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 6$: The division process in $I$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 11.3$: The division algorithm
1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.1$ Divisibility of integers
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 $2 \text{-} 2 \ (4)$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$: $\mathbf D. \, 1$
1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: $\text {(iv)}$