Answer:
Large gear: 2 revolutions, 24 teeth
Small gear: 3 revolutions, 24 teeth
Explanation:
You want the number of teeth that have touched when a 12-tooth gear and an 8-tooth gear turn until they have the same relative position as when they started.
Least common multiple
The least common multiple (LCM) of 8 and 12 is the smallest integer that is a positive integer multiple of both these numbers. That value is ...
24 = 8·3 = 12·2
It is often found quickly by making use of your knowledge of multiplication tables.
How to find it
The LCM can be found by looking at the unique factors of the numbers, taking each to the highest of the powers:
8 = 2³
12 = 2² × 3
Among the factors 2 and 3, the highest powers are 2³ and 3¹, so ...
2³ × 3¹ = 24
is the LCM.
The LCM is also the product of the numbers, divided by their greatest common factor (GCF):
LCM(8, 12) = 8·12/4 = 24
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Additional comment
The problem statement is telling you to look for the same number in the two right columns of the table. That number is 24. It goes in the blanks at the bottom.