The situation
imagine that you have three mechanical clocks in your home. (these are traditional, analog (nondigital), 12-hour clocks.) unfortunately, none of your clocks keeps time properly. each clock consistently gains or loses a fixed number of minutes each hour, but different clocks may gain or lose different amounts.

your task
at noon today, all the clocks are correct. investigate when, if ever, the clocks will be correct again. start by looking at these three specific clocks.
if a clock gains 10 minutes each hour, when will it be correct again?
if a clock loses 3 minutes each hour, when will it be correct again?
if a clock gains 7 minutes each hour, when will it be correct again?
then explore this more general situation: suppose you have three clocks in your house. the grandfather clock loses g minutes every hour, the alarm clock gains c minutes every hour, and the watch is permanently stuck at the same time. find a general rule for the next time that all three clocks will be correct.