Book VII · Proposition 22

VII.22

The least numbers of those which have the same ratio with them are prime to one another.Heath, 1908

Every step, checked

• both are numbersgiven
• the least of the ratio are prime to one anotherby VII.22

What it needs, and what needs it

Needs: nothing earlier.

Used by: VII.33 X.3

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.22",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 200), rng.randint(2, 200)),
)
def prop_VII_22(a: int, b: int) -> Out:
    hypothesis("both are numbers", a > 1 and b > 1, guard=True)
    least = least_terms(a, b)
    claim("the least of the ratio are prime to one another", "VII.22",
          coprime(*least))
    return Out(least=least)