Proposition IX.32 — Each of the numbers which are continually doubled beginning …

Proof

2^{n}

has no odd divisors greater than 1 (Euclid's lemma); hence its only factorisation is

2 \times 2^{n - 1}

, even-times-even.

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Depends on (2)

VII.30Proposition VII.30If two numbers by multiplying one another make some number, and any prime number measure the product, it will also…
VII.8Definition VII.8An even-times even number is that which is measured by an even number according to an even number.

Required by (dependents) (1)

IX.34Proposition IX.34If an even number be neither one of those which are doubled from a duad, nor have its half odd, it is both even-times…

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