§7.prop.24ἐὰν δύο ἀριθμοὶ πρός τινα ἀριθμὸν πρῶτοι ὦσιν, καὶ ὁ ἐξ αὐτῶν γενόμενος πρὸς τὸν αὐτὸν πρῶτος ἔσται.
If two numbers be prime to any number, their product also will be prime to the same.
δύο γὰρ ἀριθμοὶ οἱ Α, Β πρός τινα ἀριθμὸν τὸν Γ πρῶτοι ἔστωσαν, καὶ ὁ α τὸν Β πολλαπλασιάσας τὸν Δ ποιείτω· λέγω, ὅτι οἱ Γ, Δ πρῶτοι πρὸς ἀλλήλους εἰσίν.
Let two numbers A, B be prime to any number Γ, and let A by multiplying B make Δ; I say that Γ, Δ are prime to one another.
εἰ γὰρ μή εἰσιν οἱ Γ, Δ πρῶτοι πρὸς ἀλλήλους, μετρήσει τοὺς Γ, Δ ἀριθμός.
For if Γ, Δ be not prime to one another, some number will measure Γ, Δ.
μετρείτω, καὶ ἔστω ὁ Ε. καὶ ἐπεὶ οἱ Γ, Α πρῶτοι πρὸς ἀλλήλους εἰσίν, τὸν δὲ Γ μετρεῖ τις ἀριθμὸς ὁ Ε, οἱ Α, Ε ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν.
Let it measure them, and let it be E. And since Γ, A are prime to one another, and some number E measures Γ, therefore A, E are prime to one another.
ὁσάκις δὴ ὁ Ε τὸν Δ μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Ζ· καὶ ὁ Ζ ἄρα τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας.
So many units as there are in E's measuring Δ, let there be in Z; therefore Z also measures Δ according to the units in E.
ὁ Ε ἄρα τὸν Ζ πολλαπλασιάσας τὸν Δ πεποίηκεν.
Therefore E by multiplying Z has made Δ.
ἀλλὰ μὴν καὶ ὁ Α τὸν Β πολλαπλασιάσας τὸν Δ πεποίηκεν· ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν Ε, Ζ τῷ ἐκ τῶν Α, Β. ἐὰν δὲ ὁ ὑπὸ τῶν ἄκρων ἴσος ᾖ τῷ ὑπὸ τῶν μέσων, οἱ τέσσαρες ἀριθμοὶ ἀνάλογόν εἰσιν· ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Α, οὕτως ὁ Β πρὸς τὸν Ζ. οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα, τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον· ὁ Ε ἄρα τὸν Β μετρεῖ.
But indeed A also by multiplying B has made Δ; therefore the product of E, Z is equal to the product of A, B. But if the product of the extremes be equal to the product of the means, the four numbers are proportional; therefore, as E is to A, so is B to Z. But A, E are prime, and those which are prime are also least, and the least numbers of those which have the same ratio measure those which have the same ratio the same number of times, the greater the greater, and the less the less, that is, the antecedent the antecedent, and the consequent the consequent; therefore E measures B.
μετρεῖ δὲ καὶ τὸν Γ· ὁ Ε ἄρα τοὺς Β, Γ μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ ἐστὶν ἀδύνατον.
But it also measures Γ; therefore E measures B, Γ, which are prime to one another; which is impossible.
οὐκ ἄρα τοὺς Γ, Δ ἀριθμοὺς ἀριθμός τις μετρήσει.
Therefore no number will measure the numbers Γ, Δ.
οἱ Γ, Δ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.
Therefore Γ, Δ are prime to one another; which was to be proved.
§7.prop.25ἐὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, ὁ ἐκ τοῦ ἑνὸς αὐτῶν γενόμενος πρὸς τὸν λοιπὸν πρῶτος ἔσται.
If two numbers be prime to one another, the product of one of them will be prime to the remaining one.
ἔστωσαν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους οἱ Α, Β, καὶ ὁ Α ἑαυτὸν πολλαπλασιάσας τὸν Γ ποιείτω· λέγω, ὅτι οἱ Β, Γ πρῶτοι πρὸς ἀλλήλους εἰσίν.
Let A, B be two numbers prime to one another, and let A by multiplying itself make Γ; I say that B, Γ are prime to one another.
κείσθω γὰρ τῷ Α ἴσος ὁ Δ. ἐπεὶ οἱ Α, Β πρῶτοι πρὸς ἀλλήλους εἰσίν, ἴσος δὲ ὁ Α τῷ Δ, καὶ οἱ Δ, Β ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν.
For let Δ be placed equal to A. Since A, B are prime to one another, and A is equal to Δ, therefore Δ, B also are prime to one another.
ἑκάτερος ἄρα τῶν Δ, Α πρὸς τὸν Β πρῶτός ἐστιν· καὶ ὁ ἐκ τῶν Δ, Α ἄρα γενόμενος πρὸς τὸν Β πρῶτος ἔσται.
Therefore each of Δ, A is prime to B; therefore the product of Δ, A also will be prime to B.
ὁ δὲ ἐκ τῶν Δ, Α γενόμενος ἀριθμός ἐστιν ὁ Γ. οἱ Γ, Β ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.
But the number produced from Δ, A is Γ. Therefore Γ, B are prime to one another; which was to be proved.
§7.prop.26ἐὰν δύο ἀριθμοὶ πρὸς δύο ἀριθμοὺς ἀμφότεροι πρὸς ἑκάτερον πρῶτοι ὦσιν, καὶ οἱ ἐξ αὐτῶν γενόμενοι πρῶτοι πρὸς ἀλλήλους ἔσονται.
If two numbers be prime to two numbers, both to each, their products also will be prime to one another.
δύο γὰρ ἀριθμοὶ οἱ Α, Β πρὸς δύο ἀριθμοὺς τοὺς Γ, Δ ἀμφότεροι πρὸς ἑκάτερον πρῶτοι ἔστωσαν, καὶ ὁ μὲν Α τὸν Β πολλαπλασιάσας τὸν Ε ποιείτω, ὁ δὲ Γ τὸν Δ πολλαπλασιάσας τὸν Ζ ποιείτω· λέγω, ὅτι οἱ Ε, Ζ πρῶτοι πρὸς ἀλλήλους εἰσίν.
Let two numbers A, B be prime to two numbers Γ, Δ, both to each, and let A by multiplying B make E, and let Γ by multiplying Δ make Z; I say that E, Z are prime to one another.
ἐπεὶ γὰρ ἑκάτερος τῶν Α, Β πρὸς τὸν Γ πρῶτός ἐστιν, καὶ ὁ ἐκ τῶν Α, Β ἄρα γενόμενος πρὸς τὸν Γ πρῶτος ἔσται.
For since each of A, B is prime to Γ, therefore the product of A, B also will be prime to Γ.
ὁ δὲ ἐκ τῶν Α, Β γενόμενός ἐστιν ὁ Ε· οἱ Ε, Γ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν.
But the product of A, B is E; therefore E, Γ are prime to one another.
διὰ τὰ αὐτὰ δὴ καὶ οἱ Δ, Ε πρῶτοι πρὸς ἀλλήλους εἰσίν.
For the same reason indeed Δ, E also are prime to one another.
ἑκάτερος ἄρα τῶν Γ, Δ πρὸς τὸν Ε πρῶτός ἐστιν.
Therefore each of Γ, Δ is prime to E.
καὶ ὁ ἐκ τῶν Γ, Δ ἄρα γενόμενος πρὸς τὸν Ε πρῶτος ἔσται.
Therefore the product of Γ, Δ also will be prime to E.
ὁ δὲ ἐκ τῶν Γ, Δ γενόμενός ἐστιν ὁ Ζ. οἱ Ε, Ζ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.
But the product of Γ, Δ is Z. Therefore E, Z are prime to one another; which was to be proved.