How to determine if a statement is true

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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.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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