Book VII · Proposition 27

VII.27

If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another {and this is always the case with the extremes}.Heath, 1908

Every step, checked

• the numbers are prime to one anothergiven
• their squares are prime to one anotherby VII.25
• and so are their cubes, and this is always the case with the extremesby VII.26

What it needs, and what needs it

Needs: VII.25 VII.26

Used by: VIII.3

Depth: 2 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.27",
    THEOREM,
    sample=coprime_pair,
)
def prop_VII_27(a: int, b: int) -> Out:
    hypothesis("the numbers are prime to one another", coprime(a, b))
    because(prop_VII_25, a, b)
    because(prop_VII_26, a, a, b, b)

    claim("their squares are prime to one another", "VII.25", coprime(a * a, b * b))
    claim("and so are their cubes, and this is always the case with the extremes",
          "VII.26", coprime(a * a * a, b * b * b))
    return Out(squares=(a * a, b * b))