Book VII · Proposition 26

VII.26

If two numbers be prime to two numbers, both to each, their products also will be prime to one another.Heath, 1908

Every step, checked

• each of the first two is prime to each of the last twogiven
• the two products are prime to one anotherby VII.24

What it needs, and what needs it

Needs: VII.24

Used by: VII.27

Depth: 1 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.26",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 40), rng.randint(2, 40), rng.randint(2, 40),
                        rng.randint(2, 40)),
)
def prop_VII_26(a: int, b: int, c: int, d: int) -> Out:
    hypothesis("each of the first two is prime to each of the last two",
               coprime(a, c) and coprime(a, d) and coprime(b, c) and coprime(b, d))
    because(prop_VII_24, a, b, c)

    claim("the two products are prime to one another", "VII.24",
          coprime(a * b, c * d))
    return Out(products=(a * b, c * d))