Divisor Divides Multiple/Proof 2

Theorem

Let $a, b$ be integers.

Let:

$a \divides b$

where $\divides$ denotes divisibility.


Then:

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


Proof

\(\ds a\) \(\divides\) \(\ds b\)
\(\ds \leadsto \ \ \) \(\ds \exists x \in \Z: \, \) \(\ds b\) \(=\) \(\ds x a\) 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

$\blacksquare$

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