If matrix A has 180 elements, then the number of possible orders of A is

Options

A. 18
B. 10
C. 36
D. 35

Answer

To determine the number of possible orders of a matrix \( A \) that has 180 elements, we need to find all the pairs of positive integers \( (m, n) \) such that \( m \times n = 180 \). Here, \( m \) represents the number of rows and \( n \) represents the number of columns in the matrix.

### Step-by-Step Solution:

1. **Understand the Problem**:
We need to find pairs of integers \( (m, n) \) such that the product \( m \times n = 180 \).

2. **Find the Factors of 180**:
To find the possible orders of the matrix, we first need to determine the factors of 180. We can do this by performing prime factorization.

\[ 180 = 2^2 \times 3^2 \times 5^1 \]

3. **Calculate the Number of Factors**:
The number of factors of a number can be calculated using the formula:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]
where \( e_1, e_2, \ldots \) are the exponents in the prime factorization.

For \( 180 = 2^2 \times 3^2 \times 5^1 \):
- The exponent of 2 is 2, so \( e_1 + 1 = 2 + 1 = 3 \)
- The exponent of 3 is 2, so \( e_2 + 1 = 2 + 1 = 3 \)
- The exponent of 5 is 1, so \( e_3 + 1 = 1 + 1 = 2 \)

Therefore, the total number of factors is:
\[ 3 \times 3 \times 2 = 18 \]

4. **List the Factor Pairs**:
The pairs of factors \( (m, n) \) that multiply to give 180 are:
- \( (1, 180) \)
- \( (2, 90) \)
- \( (3, 60) \)
- \( (4, 45) \)
- \( (5, 36) \)
- \( (6, 30) \)
- \( (9, 20) \)
- \( (10, 18) \)
- \( (12, 15) \)

Each pair represents a possible order of the matrix.

5. **Conclusion**:
Since we have found 18 factors of 180, the number of possible orders of the matrix \( A \) is 18.

### Final Answer:
The number of possible orders of matrix \( A \) is **18**.

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