Common Divisor Divides Integer Combination/Proof 1
Jump to navigation
Jump to search
Corollary to Common Divisor in Integral Domain Divides Linear Combination
Let $c$ be a common divisor of two integers $a$ and $b$.
That is:
- $a, b, c \in \Z: c \divides a \land c \divides b$
Then $c$ divides any integer combination of $a$ and $b$:
- $\forall p, q \in \Z: c \divides \paren {p a + q b}$
Proof
We have that the Integers form Integral Domain.
The result then follows from Common Divisor in Integral Domain Divides Linear Combination.
$\blacksquare$