Book IX, Proposition 25

If from an even number an odd number be subtracted, the remainder will be odd.

Ἐὰν ἀπὸ ἀρτίου ἀριθμοῦ περισσὸς ἀφαιρεθῇ, ὁ λοιπὸς περισσὸς ἔσται. Ἀπὸ γὰρ ἀρτίου τοῦ ΑΒ περισσὸς ἀφῃρήσθω ὁ ΒΓ: λέγω, ὅτι ὁ λοιπὸς ὁ ΓΑ περισσός ἐστιν. Ἀφῃρήσθω γὰρ ἀπὸ τοῦ ΒΓ μονὰς ἡ ΓΔ: ὁ ΔΒ ἄρα ἄρτιός ἐστιν. ἔστι δὲ καὶ ὁ ΑΒ ἄρτιος: καὶ λοιπὸς ἄρα ὁ ΑΔ ἄρτιός ἐστιν. καί ἐστι μονὰς ἡ ΓΔ: ὁ ΓΑ ἄρα περισσός ἐστιν: ὅπερ ἔδει δεῖξαι.

If from an even number an odd number be subtracted, the remainder will be odd. For from the even number AB let the odd number BC be subtracted; I say that the remainder CA is odd. For let the unit CD be subtracted from BC; therefore DB is even. [VII. Def. 7] But AB is also even; therefore the remainder AD is also even. [IX. 24]

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