Divisor Divides Multiple

Theorem

Let $a, b$ be integers.

Let:

$a \divides b$

where $\divides$ denotes divisibility.

Then:

$\forall c \in \Z: a \divides b c$

Proof 1

Let $a \divides b$.

From Integer Divides Zero:

$a \divides 0$

Thus $a$ is a common divisor of $b$ and $0$.

From Common Divisor Divides Integer Combination:

$\forall p, q \in \Z: a \divides \paren {p \cdot b + q \cdot 0}$

Putting $p = c$ and $q = 1$ (for example):

$a \divides \paren {c b + 0}$

Hence the result.

$\blacksquare$

Proof 2

\(\ds a\)

\(\divides\)

\(\ds b\)

\(\ds \leadsto \ \ \)

\(\ds \exists x \in \Z: \, \)

\(\ds b\)

\(=\)

\(\ds x a\)

Definition of Divisor of Integer

\(\ds \leadsto \ \ \)

\(\ds b c\)

\(=\)

\(\ds x c a\)

\(\ds \leadsto \ \ \)

\(\ds \exists z \in \Z: \, \)

\(\ds b c\)

\(=\)

\(\ds z a\)

where $z = x c$

\(\ds \leadsto \ \ \)

\(\ds a\)

\(\divides\)

\(\ds b c\)

Definition of Divisor of Integer

$\blacksquare$

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: Problems $2.2$: $2 \ \text{(a)}$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$: $\mathbf D \, 5$
• 1982: Martin Davis: Computability and Unsolvability (2nd ed.) ... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers: Corollary $1$