Period of sum of two periodic sequences
If you sum a sequence with period $a$ and a sequence with period $b$, then you get a sequence which is $\mathsf{LCM}(a,b)$-periodic. But the new sequence might have a smaller period, as the following example demonstrates (addition is modulo 10):

022441133502244113350224411335
000772999611888000772999611888
022113022113022113022113022113


The first sequence has period 10 (0224411335), the second sequence as period 15 (000772999611888), and the sum sequence has period 6 (022113), although the LCM is 30.

More generally, the sum of a sequence of period $i$ and a sequence of period $j$ could have period exactly $k$ iff $\mathsf{LCM}(i,j) = \mathsf{LCM}(i,k) = \mathsf{LCM}(j,k)$. See Restrepo and Chacon, On the period of sums of discrete periodic signals.