2140.31 – Gear Wheels

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.