Book VII · Proposition 21

VII.21

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

Every step, checked

• the numbers are prime to one anothergiven
• they are already in least termsby VII.21
• so no smaller pair has the same ratioby VII.20

What it needs, and what needs it

Needs: VII.20

Used by: VII.33 VIII.1 VIII.3

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.21",
    THEOREM,
    sample=coprime_pair,
)
def prop_VII_21(a: int, b: int) -> Out:
    hypothesis("the numbers are prime to one another", coprime(a, b))
    because(prop_VII_20, a, b, a, b)

    claim("they are already in least terms", "VII.21", least_terms(a, b) == (a, b))
    claim("so no smaller pair has the same ratio", "VII.20",
          not any(x * b == y * a for x in range(1, a) for y in range(1, b)))
    return Out()