Proof of Even/Odd Case for a|bc in Z: Does a|b or a|c?

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The discussion centers on the mathematical statement that for all integers \( a, b, c \in \mathbb{Z} \), if \( a \) divides \( bc \), then \( a \) must divide either \( b \) or \( c \). Participants assert that this statement is false and emphasize the necessity of finding a counterexample to demonstrate its invalidity. The conversation highlights the importance of understanding the properties of even and odd integers in relation to divisibility.

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\forall a,b,c \in Z a|bc \Rightarrow a|b or a|c

To prove this do I only need to show that 'a' can be either even or odd?
 
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John112 said:
\forall a,b,c \in Z a|bc \Rightarrow a|b or a|c

To prove this do I only need to show that 'a' can be either even or odd?

You can't prove it because it isn't true. Find a counterexample.
 

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