Common Divisor Divides Integer Combination

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 $b$:

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

Corollary

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

General Result

Let $c$ be a common divisor of a set of integers $A := \set {a_1, a_2, \dotsc, a_n}$.

That is:

$\forall x \in A: c \divides x$

Then $c$ divides any integer combination of elements of $A$:

$\forall x_1, x_2, \dotsc, x_n \in \Z: c \divides \paren {a_1 x_2 + a_2 x_2 + \dotsb + a_n x_n}$

Proof 1

We have that the Integers form Integral Domain.

The result then follows from Common Divisor in Integral Domain Divides Linear Combination.

$\blacksquare$

Proof 2

$\ds c$

$\divides$

$\ds a$

$\ds \leadsto \ \ $

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

$\ds a$

$=$

$\ds x c$

Definition of Divisor of Integer

$\ds c$

$\divides$

$\ds b$

$\ds \leadsto \ \ $

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

$\ds b$

$=$

$\ds y c$

Definition of Divisor of Integer

$\ds \leadsto \ \ $

$\ds \forall p, q \in \Z: \, $

$\ds p a + q b$

$=$

$\ds p x c + q y c$

Substitution for $a$ and $b$

$\ds $

$=$

$\ds \paren {p x + q y} c$

Integer Multiplication Distributes over Addition

$\ds \leadsto \ \ $

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

$\ds p a + q b$

$=$

$\ds z c$

where $z = p x + q y$

$\ds \leadsto \ \ $

$\ds c$

$\divides$

$\ds \paren {p a + q b}$

Definition of Divisor of Integer

$\blacksquare$

Sources

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 $2 \text{-} 2 \ (7)$
1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: $\text {(v)}$