How to prove that every set of two or more numbers has a common multiple or factor?

Question

I want to prove this:

Every combination of two or more real numbers always has a common multiple or common factor.

I'm unsure if this applies only to nonzero integers, or real numbers in general.

Answer 1

Every real number is a multiple of every other real number (except zero). For instance $2.3$ is a multiple of $\sqrt{7}$ since

$$2.3 = \sqrt{7}\cdot\dfrac{2.3}{\sqrt{7}}$$

Similarly, any real number, except $0$, is a common factor of any other real number.

All integers do have a common multiple, their product. That is $a\cdot b$ is a common multiple of both $a$ and $b$. So, for example a common multiple of $2$ and $31$ is $2\cdot 31=62$. Similarly, a common multiple of $a,b$ and $c$ is $a\cdot b\cdot c$.

All pairs of integers have a common factor as well, $1$ (as well as $-1$). Certain pairs of integers only have these as common factors. We say they are coprime or relatively prime. For example, $2$ and $9$ are coprime, but $10$ and $25$ are not, as $5$ is a common factor.