Suppose we consider the decimal representation of rational numbers less than 1 and consider them as a sequence. Now consider all such rational numbers where it is known that the period for each of the sequence is bounded by some number $M$. Now I construct a new sequence where the $n$-th digit is $2$ if the $n$-th digit in the $n$-th rational number in the original sequence is not equal to 2 and is 3 if the $n$-th digit in the $n$-th rational number in the original sequence is equal to 2. Now is it correct that the period of the new sequence is the least common multiple of the periods of the original sequence ? How to prove it ?
Can the new sequence be periodic for some other way of constructing the new number ? Or, it can be proved that it can never be periodic. Because each time I am taking just one element from the original sequences whether the original sequence is periodic or not that does not make any difference in the new sequence. Is this the reason ?