Book VII · Proposition 23

VII.23

If two number be prime to one another, the number which measures the one of them will be prime to the remaining number.Heath, 1908

Every step, checked

• the numbers are prime to one anothergiven
• every measure of the one is prime to the otherby VII.23

What it needs, and what needs it

Needs: nothing earlier.

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.23",
    THEOREM,
    sample=lambda rng: coprime_pair(rng),
)
def prop_VII_23(a: int, b: int) -> Out:
    hypothesis("the numbers are prime to one another", coprime(a, b))
    divisors = [d for d in range(2, a + 1) if measures(d, a)]
    claim("every measure of the one is prime to the other", "VII.23",
          all(coprime(d, b) for d in divisors))
    return Out(divisors=divisors)