Book IX · Proposition 29

IX.29

If an odd number by multiplying an odd number make some number, the product will be odd.Heath, 1908

Every step, checked

• an odd number multiplies an odd onegiven
• the product is oddby IX.23

What it needs, and what needs it

Needs: IX.23

Rests on: Def.VII.6

Depth: 3 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("IX.29", THEOREM,
             sample=lambda rng: _pair_by_parity(rng, False, False))
def prop_IX_29(a: int, b: int) -> Out:
    hypothesis("an odd number multiplies an odd one",
               not measures(2, a) and not measures(2, b))
    because(prop_IX_23, [a, b, a]) if all(not measures(2, n) for n in (a, b)) else None

    claim("the product is odd", "IX.23", not measures(2, a * b))
    return Out(product=a * b)