Book VII · Proposition 24

VII.24

If two numbers be prime to any number, their product also will be prime to the same.Heath, 1908

Every step, checked

• both are prime to the thirdgiven
• their product is prime to it as wellby VII.24

What it needs, and what needs it

Needs: nothing earlier.

Used by: VII.25 VII.26

Depth: 0 steps of argument above the first principles. Parallel postulate: not needed.

What it takes on trust

Nothing. It draws no intersections and reads nothing off the picture.

The proposition as code

@proposition(
    "VII.24",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 60), rng.randint(2, 60), rng.randint(2, 60)),
)
def prop_VII_24(a: int, b: int, c: int) -> Out:
    hypothesis("both are prime to the third", coprime(a, c) and coprime(b, c))
    claim("their product is prime to it as well", "VII.24", coprime(a * b, c))
    return Out(product=a * b)