It is known that a string $s$ is actually made up of repetitions of another string $s_1$ of length $L_1$.
Also $s$ can be thought of as made up of repetitions of another string $s_2$ of length $L_2$.
For example the string $s = abababab$ is made by repeating the substring $s_1$ "$ab$" of length $L_1= 2$ or by repeating the substring $s_2$ "$abab$" of length $L_2=4$.
I believe that in such a case $s$ can be made by repeating substring of length $\gcd(L_1,L_2)$. For example if $L_1=6$ and $L_2=10$ then $s$ should be repetitive in a substring of length $2$ (= $\gcd(6,10)$) too. This seems intuitive but how do I show this formally ?