A related technique is to use distinct prime numbers for periodicities in the first place:

http://www.stdlib.net/~colmmacc/2009/11/26/period-pain-3/

(http://www.stdlib.net/~colmmacc/2009/09/27/period-pain-part-... http://www.stdlib.net/~colmmacc/2009/09/14/period-pain/ for the background material).

The reasons primes are optimal should be obvious :)