## Book IX, Proposition 31

If an odd number be prime to any number, it will also be prime to the double of it.

Ἐὰν περισσὸς ἀριθμὸς πρός τινα ἀριθμὸν πρῶτος ᾖ, καὶ πρὸς τὸν διπλασίονα αὐτοῦ πρῶτος ἔσται. Περισσὸς γὰρ ἀριθμὸς ὁ Α πρός τινα ἀριθμὸν τὸν Β πρῶτος ἔστω, τοῦ δὲ Β διπλασίων ἔστω ὁ Γ: λέγω, ὅτι ὁ Α [ καὶ ] πρὸς τὸν Γ πρῶτός ἐστιν. Εἰ γὰρ μή εἰσιν [ οἱ Α, Γ ] πρῶτοι, μετρήσει τις αὐτοὺς ἀριθμός. μετρείτω, καὶ ἔστω ὁ Δ. καί ἐστιν ὁ Α περισσός: περισσὸς ἄρα καὶ ὁ Δ. καὶ ἐπεὶ ὁ Δ περισσὸς ὢν τὸν Γ μετρεῖ, καί ἐστιν ὁ Γ ἄρτιος, καὶ τὸν ἥμισυν ἄρα τοῦ Γ μετρήσει [ ὁ Δ ]. τοῦ δὲ Γ ἥμισύ ἐστιν ὁ Β: ὁ Δ ἄρα τὸν Β μετρεῖ. μετρεῖ δὲ καὶ τὸν Α. ὁ Δ ἄρα τοὺς Α, Β μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους: ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα ὁ Α πρὸς τὸν Γ πρῶτος οὔκ ἐστιν. οἱ Α, Γ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν: ὅπερ ἔδει δεῖξαι. | If an odd number be prime to any number, it will also be prime to the double of it. For let the odd number A be prime to any number B, and let C be double of B; I say that A is prime to C. For, if they are not prime to one another, some number will measure them. Let a number measure them, and let it be D. Now A is odd; therefore D is also odd. And since D which is odd measures C, and C is even, therefore [D] will measure the half of C also. [IX. 30] But B is half of C; therefore D measures B. But it also measures A; therefore D measures A, B which are prime to one another: which is impossible. Therefore A cannot but be prime to C. |