Book IX, Proposition 26

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

Ἐὰν ἀπὸ περισσοῦ ἀριθμοῦ περισσὸς ἀφαιρεθῇ, ὁ λοιπὸς ἄρτιος ἔσται. Ἀπὸ γὰρ περισσοῦ τοῦ ΑΒ περισσὸς ἀφῃρήσθω ὁ ΒΓ: λέγω, ὅτι ὁ λοιπὸς ὁ ΓΑ ἄρτιός ἐστιν. Ἐπεὶ γὰρ ὁ ΑΒ περισσός ἐστιν, ἀφῃρήσθω μονὰς ἡ ΒΔ: λοιπὸς ἄρα ὁ ΑΔ ἄρτιός ἐστιν. διὰ τὰ αὐτὰ δὴ καὶ ὁ ΓΔ ἄρτιός ἐστιν: ὥστε καὶ λοιπὸς ὁ ΓΑ ἄρτιός ἐστιν: ὅπερ ἔδει δεῖξαι.

If from an odd number an odd number be subtracted, the remainder will be even. For from the odd number AB let the odd number BC be subtracted; I say that the remainder CA is even. For, since AB is odd, let the unit BD be subtracted; therefore the remainder AD is even. [VII. Def. 7]

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