The marks on the faces of these gears are aligned as shown. What is the least number of turns that gear A must make so that the marks are all re-aligned in their original position? (Note: Gear B has two options for "original". Either one is OK.)
Solution
Let's define "turn" as a complete turn, and let's define "click" as a partial turn of one notch on a wheel.
A will return to its original position after any multiple of four clicks.
B will return to its original position after any multiple of three clicks (see the note in the statement of the problem).
C will return to its original position after any multiple of five clicks.
D will return to its original position after any multiple of four clicks (same as A).
The least common multiple of these click numbers is 60. If A makes 60 clicks it will be in its original position after 15 turns. B will be in its "original" position after 20 turns. C will be in its original position after 12 turns. And D will be in its original position after 15 turns.
So we see that If A makes 15 turns everything will be fine.