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 find the number of possible orders of a matrix \( A \) that has 180 elements, we need to consider the different pairs of integers \( (m, n) \) such that \( m \times n = 180 \), where \( m \) is the number of rows and \( n \) is the number of columns.

### Step-by-Step Solution:

1. **Understand the relationship**: The total number of elements in a matrix is given by the product of its number of rows and columns. Therefore, for matrix \( A \), we have:
\[ m \times n = 180 \]

2. **Find the factors of 180**: To determine the possible orders of the matrix, we need to find all pairs of factors of 180. We start by finding the prime factorization of 180:
\[ 180 = 2^2 \times 3^3 \times 5^1 \]

3. **List the factors**: The factors of 180 can be found by considering all combinations of the prime factors. The factors of 180 are:
\[ 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 \]

4. **Form pairs of factors**: For each factor \( m \), we can find \( n \) by calculating \( n = \frac{180}{m} \). The pairs \( (m, n) \) are:
- \( (1, 180) \)
- \( (2, 90) \)
- \( (3, 60) \)
- \( (4, 45) \)
- \( (5, 36) \)
- \( (6, 30) \)
- \( (9, 20) \)
- \( (10, 18) \)
- \( (12, 15) \)

5. **Count the unique pairs**: Each pair \( (m, n) \) is considered unique, and we also consider the reverse pairs \( (n, m) \) as valid orders. Therefore, we have:
- \( (1, 180) \) and \( (180, 1) \)
- \( (2, 90) \) and \( (90, 2) \)
- \( (3, 60) \) and \( (60, 3) \)
- \( (4, 45) \) and \( (45, 4) \)
- \( (5, 36) \) and \( (36, 5) \)
- \( (6, 30) \) and \( (30, 6) \)
- \( (9, 20) \) and \( (20, 9) \)
- \( (10, 18) \) and \( (18, 10) \)
- \( (12, 15) \) and \( (15, 12) \)

6. **Total number of unique orders**: Since each of the pairs can be arranged in two ways (i.e., \( (m, n) \) and \( (n, m) \)), we count each pair only once. The total number of unique pairs is:
- \( 9 \) unique pairs.

Thus, the total number of possible orders of matrix \( A \) is \( 18 \).

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