Book IX, Proposition 27

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

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

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

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