Periodicity of function - finding the period

Feb 2013
If \(\displaystyle f_1(x), f_2(x), f_3(x)\) are periodic functions with period \(\displaystyle T_1,T_2,T_3\) respectively then, we have
h(x) =\(\displaystyle af_1(x) \pm bf_2(x) \pm cf_3(x)\) has period as,

L.C.M of \(\displaystyle (T_1,T_2,T_3)\) ; if h(x) is not an even function

\(\displaystyle \frac{1}{2}\)L.C.M of (\(\displaystyle T_1,T_2,T_3)\) ; if h(x) is even function

L.C.M of \(\displaystyle (\frac{a}{b},\frac{c}{d},\frac{e}{f}) \Rightarrow \frac{L.C.M of (a,c,e)}{H.C.F. of (b,d,f)}\)

Example : Find period of f(x) =tan3x +sin\(\displaystyle \frac{x}{3}\)

Period of tan3x is \(\displaystyle |\frac{\pi}{3}|\) and period of \(\displaystyle \sin\frac{x}{3}\) is \(\displaystyle |2\pi \times \frac{3}{1}|\) = \(\displaystyle |6\pi|\). Thus f(x) is periodic with period \(\displaystyle |6\pi|.\)

My question is why we have to take L.C.M and H.C.F to find the period of function. Can you guide me or give me source from where this derivation arrived?