A regular polygon is a plane shape that is enclosed by 3 or more lines of equal lengths and it is always symmetric having all the interior angles equal to each other, examples include equilateral triangle, square, regular pentagon, regular haxagon and so on.
Verma has 120 identical matchsticks and he wants to form a regular polygon, how many distinct polygons can he create using all the matchsticks together?

Options

A. 10
B. 12
C. 14
D. 16

Answer

To determine how many distinct regular polygons Verma can create using all 120 matchsticks, we need to analyze the properties of regular polygons and how the matchsticks can be arranged.

### Step-by-Step Solution:

1. **Understanding Regular Polygons**: A regular polygon has equal sides and equal angles. The minimum number of sides for a polygon is 3 (triangle).

2. **Forming Polygons with Matchsticks**: Since each side of the polygon requires one matchstick, the total number of sides (n) of the polygon must be a divisor of the total number of matchsticks (120).

3. **Finding Divisors of 120**: We need to find all divisors of 120 that are greater than or equal to 3 (since a polygon must have at least 3 sides).

- The prime factorization of 120 is:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]
- To find the divisors, we can calculate:
- The total number of divisors can be found using the formula:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \]
where \(e_1, e_2, e_3\) are the powers of the prime factors. For 120, this gives:
\[ (3 + 1)(1 + 1)(1 + 1) = 4 \times 2 \times 2 = 16 \]
- The divisors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.

4. **Filtering Valid Polygons**: We need to filter these divisors to find those that are 3 or greater:
- Valid divisors (sides of polygons): 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.

5. **Counting Distinct Polygons**: Now we count the valid divisors:
- The valid divisors that correspond to regular polygons are: 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.
- This gives us a total of 14 distinct regular polygons.

### Conclusion:
Verma can create **14 distinct regular polygons** using all 120 matchsticks.

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