Why the resultant period of two signals is the LCM of the individual period of that signal?
Take two sine waves of periods \$T_1 = 2s\$ and \$T_2 = 3s\$. 

Suppose they both start at time = 0s.

Then their "end of the cycle" points coincide only at the multiples of \$LCM(T_1,T_2) = LCM(2,3) =  6s \$

While adding both signals, we take summation of all respective points of both signals at a time instant, and we get a set of non-repeating values between 0 to 6s interval. After that, these values roll over and repeat every 6s. Hence the period of the resulting signal becomes 6s.