If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

Explanation

To determine the possible orders of a matrix based on the number of elements it contains, we need to find pairs of positive integers (rows and columns) whose product equals the total number of elements.

### Step-by-Step Solution:

1. **Understanding the Order of a Matrix:**
The order of a matrix is defined as the number of rows multiplied by the number of columns. If a matrix has \( n \) elements, then the order can be represented as \( r \times c \) where \( r \) is the number of rows and \( c \) is the number of columns.

2. **Finding Possible Orders for 24 Elements:**
We need to find pairs of integers \( (r, c) \) such that \( r \times c = 24 \).

- The factors of 24 are:
- \( 1 \times 24 \)
- \( 2 \times 12 \)
- \( 3 \times 8 \)
- \( 4 \times 6 \)
- \( 6 \times 4 \)
- \( 8 \times 3 \)
- \( 12 \times 2 \)
- \( 24 \times 1 \)

Thus, the possible orders for a matrix with 24 elements are:
- \( 1 \times 24 \)
- \( 2 \times 12 \)
- \( 3 \times 8 \)
- \( 4 \times 6 \)
- \( 6 \times 4 \)
- \( 8 \times 3 \)
- \( 12 \times 2 \)
- \( 24 \times 1 \)

3. **Finding Possible Orders for 13 Elements:**
Now, we need to find pairs of integers \( (r, c) \) such that \( r \times c = 13 \).

- Since 13 is a prime number, its only factors are:
- \( 1 \times 13 \)
- \( 13 \times 1 \)

Therefore, the possible orders for a matrix with 13 elements are:
- \( 1 \times 13 \)
- \( 13 \times 1 \)

### Summary of Possible Orders:
- For 24 elements: \( 1 \times 24, 2 \times 12, 3 \times 8, 4 \times 6, 6 \times 4, 8 \times 3, 12 \times 2, 24 \times 1 \) (Total: 8 orders)
- For 13 elements: \( 1 \times 13, 13 \times 1 \) (Total: 2 orders)