$1\mathrm{.}14.$ Let $x_1(t)$ and $x_2(t)$ be periodic signals with fundamental periods $T_1$ and $T_2,$ respectively. Under what conditions is the sum $x(t)=x_1(t)+x_2(t)$ periodic, and what is the fundamental period of $x(t)$ if it is periodic?

Since $x_1(t)$ and $x_2(t)$ are periodic with fundamental periods $T_1$ and $T_2,$ respectively, we have

$x_1(t)=x_1(t+T_1)=x_1(t+m{T}_1),m=$ positive integer

$x_2(t)=x_2(t+T_2)=x_2(t+k{T}_2),k=$ positive integer

Thus,

$x(t)=x_1(t+m{T}_1)+x_2(t+k{T}_2)$

In order for $x(t)$ to be periodic with period $T,$ one needs

$x(t+T)=x_1(t+T)+x_2(t+T)=x_1(t+m{T}_1)+x_2(t+k{T}_2)$

Thus, we must have

$m{T}_1=k{T}_2=T$

or

$\frac{T_1}{T_2}=\frac{k}{m}=$ rational number

In other words, the sum of two periodic signals is periodic only if the ratio of their respective periods can be expressed as a rational number. Then the fundamental period is the least