Book VII · Proposition 25

VII.25

If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.Heath, 1908

Every step, checked

• the numbers are prime to one anothergiven
• the square of the one is prime to the otherby 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.25",
    THEOREM,
    sample=coprime_pair,
)
def prop_VII_25(a: int, b: int) -> Out:
    hypothesis("the numbers are prime to one another", coprime(a, b))
    because(prop_VII_24, a, a, b) if coprime(a, b) else None

    claim("the square of the one is prime to the other", "VII.24", coprime(a * a, b))
    return Out(square=a * a)