Given two functions:
$y_1 = \sin{2x}$
$y_2 = \sin{3x}$
Is the point at which they both go towards a peak the lowest common multiple of the 2 and 3 (so 6)?
If so is this the case for all period functions? I am struggling to think of something that would disprove the above.
The idea is right but the application needs some work. The period of $\sin (2x)$ is $\pi$ and the period of $\sin (3x)$ is $\frac {2\pi}3$. The common period of the two is smallest number that is an integer multiple of both of them, here $2\pi$. Given two functions with periods $p,q$ you can find the common period by dividing $\frac qp$. If this is not rational, there is no common period. If it is, express it in lowest terms and the common period is the denominator times the period $q$.