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State/Two out of three principle for divisibility
From SlugmathWiki
Proposition: (Two out of three principle for divisibility.) Suppose that $x,y,z \in \ZZ$, and $d \in \ZZ$, and $x + y = z$ . Then, if $d$ divides two elements of the set $\{x,y,z \}$, then $d$ divides all three elements of the set $\{x,y,z \}$.
More generally, suppose that $R$ is a commutative ring. Suppose that $x,y,z \in R$, and $d \in R$, and $x + y = z$. Then, if $d$ divides two elements of the set $\{x,y,z \}$, then $d$ divides all three elements of the set $\{ x, y, z \}$.
Logical Connections
This statement logically relies on the following definitions and statements: Def/Divides, Def/Distributive
The following statements and definitions rely on the material of this page: State/Greatest common divisors can be found with the Euclidean algorithm, State/Linear Diophantine equations can be solved with the Euclidean algorithm, State/Primes dividing a product must divide a factor, State/Systems of two linear Diophantine equations, and State/There are infinitely many prime numbers
To visualize the logical connections between this statements and other items of mathematical knowledge, you can visit the following cluster(s), and click the "Visualize" tab: Clust/Basic number theory, Clust/Basic ring theory
