POTW #29 - Necklace Patterns

POTW #28 ignited some debate in class as to how to most effectively answer it (and actually, whether it could be answered at all). Please see two possible ways to solve this surprisingly challenging question. The first soliution deals more with factors/mutliples and number sense in general. The second solution uses algebra.
POTW #29 is below: Some patterning and number sense connections on this one!