PROPOSITION 25.

If two numbers be prime to one another, the product of one of them into itself[1] will be prime to the remaining one.

Let A, B be two numbers prime to one another, and let A by multiplying itself make C:

I say that B, C are prime to one another.

A D B C

For let D be made equal to A.

Since A, B are prime to one another, and A is equal to D,
therefore D, B are also prime to one another.
Therefore each of the two numbers D, A is prime to B ;
[VII. 24]therefore the product of D, A will also be prime to B.
But the number which is the product of D, A is C.
Therefore C, B are prime to one another.
Q. E. D.

Dependency Graph

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"VII.def.12" -> "VII.def.1"; "VII.def.15" -> "VII.def.1"; "VII.1" -> "VII.def.1"; "VII.15" -> "VII.def.1"; "VII.def.3" -> "VII.def.2"; "VII.def.5" -> "VII.def.2"; "VII.def.15" -> "VII.def.2"; "VII.1" -> "VII.def.2"; "VII.2" -> "VII.def.2"; "VII.4" -> "VII.def.2"; "VII.5" -> "VII.def.2"; "VII.6" -> "VII.def.2"; "VII.9" -> "VII.def.2"; "VII.10" -> "VII.def.2"; "VII.12" -> "VII.def.2"; "VII.13" -> "VII.def.2"; "VII.15" -> "VII.def.2"; "VII.16" -> "VII.def.2"; "VII.17" -> "VII.def.2"; "VII.18" -> "VII.def.2"; "VII.19" -> "VII.def.2"; "VII.20" -> "VII.def.2"; "VII.21" -> "VII.def.2"; "VII.23" -> "VII.def.2"; "VII.24" -> "VII.def.2"; "VII.25" -> "VII.def.2"; "VII.def.20" -> "VII.def.3"; "VII.4" -> "VII.def.3"; "VII.5" -> "VII.def.3"; "VII.9" -> "VII.def.3"; "VII.10" -> "VII.def.3"; "VII.def.20" -> "VII.def.4"; "VII.4" -> "VII.def.4"; "VII.6" -> "VII.def.4"; "VII.9" -> "VII.def.4"; "VII.10" -> "VII.def.4"; "VII.def.20" -> "VII.def.5"; "VII.5" -> "VII.def.5"; "VII.1" -> "VII.def.12"; "VII.2" -> "VII.def.12"; "VII.21" -> "VII.def.12"; "VII.23" -> "VII.def.12"; "VII.24" -> "VII.def.12"; "VII.25" -> "VII.def.12"; "VII.16" -> "VII.def.15"; "VII.17" -> "VII.def.15"; "VII.18" -> "VII.def.15"; "VII.24" -> "VII.def.15"; "VII.12" -> "VII.def.20"; "VII.13" -> "VII.def.20"; "VII.17" -> "VII.def.20"; "VII.20" -> "VII.def.20"; "VII.2" -> "VII.1"; "VII.4" -> "VII.2"; "VII.20" -> "VII.4"; "VII.6" -> "VII.5"; "VII.9" -> "VII.5"; "VII.10" -> "VII.5"; "VII.12" -> "VII.5"; "VII.9" -> "VII.6"; "VII.10" -> "VII.6"; "VII.12" -> "VII.6"; "VII.10" -> "VII.9"; "VII.13" -> "VII.10"; "VII.15" -> "VII.12"; "VII.20" -> "VII.12"; "VII.17" -> "VII.13"; "VII.20" -> "VII.13"; "VII.16" -> "VII.15"; "VII.18" -> "VII.16"; "VII.21" -> "VII.16"; "VII.24" -> "VII.16"; "VII.18" -> "VII.17"; "VII.19" -> "VII.17"; "VII.19" -> "VII.18"; "VII.24" -> "VII.19"; "VII.21" -> "VII.20"; "VII.24" -> "VII.20"; "VII.24" -> "VII.21"; "VII.24" -> "VII.23"; "VII.25" -> "VII.24"; }

Required for

VII.27, VIII.2, VIII.3, VIII.6, VIII.7, VIII.8, VIII.9, VIII.14, VIII.15, VIII.16, VIII.17, VIII.21, VIII.23, VIII.24, VIII.25, VIII.26, VIII.27, IX.1, IX.2, IX.3, IX.4, IX.5, IX.6, IX.8, IX.9, IX.10, IX.12, IX.13, IX.15, IX.32, IX.36, X.9, X.10, X.29, X.30, X.31, X.32, X.33, X.34, X.35, X.39, X.40, X.41, X.45, X.46, X.48, X.49, X.50, X.51, X.52, X.53, X.57, X.58, X.59, X.60, X.63, X.64, X.65, X.68, X.69, X.70, X.71, X.72, X.76, X.77, X.78, X.82, X.83, X.84, X.85, X.86, X.87, X.88, X.89, X.90, X.91, X.94, X.95, X.96, X.100, X.101, X.102, X.105, X.106, X.107, X.108, X.109, X.110, X.111, XIII.6, XIII.11, XIII.16, XIII.17, XIII.18

See Commentary on Proposition VII.25.