Stuck on proving that this function\signal is not periodic

[Roughmar]

First of all, HI! This is my first post and my first day in this forum. =)

I am having quite a problem demystifying this function. It's on a book I have and it clearly states that it isn't periodical. I can't reach that conclusion and was hoping you could help me out.

So, the function in question is $$x(t)=sin(\pi t)cos(10t)$$

What I did was to deconstruct this into
$$\frac{1}{2}\left[ sin(\pi t+10t)-sin(\pi t-10t)\right]$$

Now, I think that the period of the first sin is $$\frac{2\pi}{\pi +10}$$ and the one from the second sin is $$\frac{2\pi}{\pi -10}$$.

I then try to find the fundamental period of the whole function:

$$n\frac{2\pi}{\pi +10}=m\frac{2\pi}{\pi -10}\Rightarrow m=\pi -10$$ and $$n=\pi+10 \Rightarrow T=2\pi$$


I know I have to be doing something wrong and possibly it's also really basic, but I got stuck.
Can anyone help me out? =)

 

[LCKurtz]

But your m and n aren't integers. Try an indirect argument. Suppose your x(t) has period P, so x(t+P) = x(t). What does that give you? What happens if you put t = 0 in it?

 

[Roughmar]

But your m and n aren't integers. Try an indirect argument. Suppose your x(t) has period P, so x(t+P) = x(t). What does that give you? What happens if you put t = 0 in it?

I had done the substitution before, and noticed it didn't work, but couldn't understand why.
As soon as you said "integer" however, I just facepalmed myself.

Thank you so much.