How to determine if a statement is true

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The statement "If n is divisible by 6, then n is divisible by 9" is false, as demonstrated by counterexamples like 6, 12, and 30, which are divisible by 6 but not by 9. A single counterexample is sufficient to prove a statement false, as a true statement cannot have any counterexamples. The discussion emphasizes the importance of counterexamples in validating mathematical assertions. Understanding this concept clarifies how to assess the truth of similar statements. The conclusion reinforces the idea that counterexamples are key to disproving false statements.
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Homework Statement


Determine if the statement is true, n is a fixed positive integer.

If n is divisible by 6, then n is divisible by 9.

Homework Equations

The Attempt at a Solution


I know this statement is false because 6 is divisible by 6, however not divisible by 9. I showed a counterexample, is that enough to demonstrate that it is a false statement?
 
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Yes. ##12## is another counterexample as is ##30##. There are many.
 
LCKurtz said:
Yes. ##12## is another counterexample as is ##30##. There are many.
Thank you very much.
 
ver_mathstats said:

Homework Statement


Determine if the statement is true, n is a fixed positive integer.

If n is divisible by 6, then n is divisible by 9.

Homework Equations

The Attempt at a Solution


I know this statement is false because 6 is divisible by 6, however not divisible by 9. I showed a counterexample, is that enough to demonstrate that it is a false statement?
Yes, any counterexample will do, as it shows the claimed statement to be false. A true statement cannot, ever, have any counterexamples!
 
Ray Vickson said:
Yes, any counterexample will do, as it shows the claimed statement to be false. A true statement cannot, ever, have any counterexamples!

Okay thank you very much, I understand now.
 

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