Set of Common Divisors of Integers is not Empty
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Theorem
Let $a, b \in \Z$ be integers.
Let $S$ be the set of common divisors of $a$ and $b$.
Then $S$ is not empty.
Proof
From One is Common Divisor of Integers:
- $1$ is a common divisor of $a$ and $b$.
Thus, whatever $a$ and $b$ are:
- $1 \in S$
The result follows by definition of empty set.
$\blacksquare$
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