Divisor Divides Multiple/Proof 2
Jump to navigation
Jump to search
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 |
$\blacksquare$