Which of the following numbers will have an even number of factors? Options are (a) 16900 (b) 52900 (c) 30000 (d) 36100.

Options

A. 16900
B. 52900
C. 30000
D. 36100

Answer

To determine which of the given numbers has an even number of factors, we need to understand the relationship between a number's prime factorization and its factors. A number will have an even number of factors if it is not a perfect square. This is because perfect squares have an odd number of factors due to the middle factor being repeated.

### Step-by-Step Solution:

1. **Identify the Numbers**: The numbers given are:
- (a) 16900
- (b) 52900
- (c) 30000
- (d) 36100

2. **Check if Each Number is a Perfect Square**:
- A perfect square has an odd number of factors.
- If a number is not a perfect square, it will have an even number of factors.

3. **Factorization of Each Number**:
- **(a) 16900**:
- \( 16900 = 169 \times 100 = 13^2 \times 10^2 = 13^2 \times (2 \times 5)^2 = 13^2 \times 2^2 \times 5^2 \)
- This is a perfect square (since \( 13^2, 2^2, \) and \( 5^2 \) are all perfect squares).

- **(b) 52900**:
- \( 52900 = 529 \times 100 = 23^2 \times 10^2 = 23^2 \times (2 \times 5)^2 = 23^2 \times 2^2 \times 5^2 \)
- This is also a perfect square.

- **(c) 30000**:
- \( 30000 = 300 \times 100 = 3 \times 100 \times 100 = 3^1 \times (10^2) \times (10^2) = 3^1 \times (2 \times 5)^4 = 3^1 \times 2^4 \times 5^4 \)
- This is not a perfect square because the exponent of 3 is odd.

- **(d) 36100**:
- \( 36100 = 361 \times 100 = 19^2 \times 10^2 = 19^2 \times (2 \times 5)^2 = 19^2 \times 2^2 \times 5^2 \)
- This is a perfect square.

4. **Conclusion**:
- The only number that is **not** a perfect square is **30000**.
- Therefore, **30000** will have an even number of factors.

### Final Answer:
(c) 30000