$$180-A=\frac{360}{n}$$

The measure $A$ in degrees, of an interior angle of a regular polygon is related to the number of sides, $n$, of the polygon by the formula above. If the measure of an interior angle of a regular polygon is less than $120\degree$, what is the greatest number of sides it can have?

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We are told the interior angle $A$ is less than $40 \degree$:

$$A \lt 120 \degree$$

We can use the given equation. Solving for $A$ allows us to substitute it into the inequality.

$$180-A=\frac{360}{n}$$ $$A=180-\frac{360}{n}$$ $$180-\frac{360}{n} \lt 120$$

Solving for n:

$$\frac{360}{n} \gt 60$$ $$360 \gt 60n$$ $$n \lt 6$$ $$n=\boxed{5}$$