How many ordered pairs of (m,n) integers satisfy `(m)/(12)=(12)/(n)`?

Options

A. `30`
B. `15`
C. `12`
D. `10`

Explanation

To solve the equation \(\frac{m}{12} = \frac{12}{n}\) for ordered pairs of integers \((m, n)\), we can follow these steps:

### Step 1: Rearrange the equation
Start by cross-multiplying to eliminate the fractions:
\[ m \cdot n = 12 \cdot 12 \]
This simplifies to:
\[ m \cdot n = 144 \]

### Step 2: Factor 144
Next, we need to find the integer pairs \((m, n)\) such that their product is 144. First, we factor 144 into its prime factors:
\[ 144 = 12 \times 12 = 2^4 \times 3^2 \]

### Step 3: Find the number of divisors
To find the number of ordered pairs \((m, n)\), we first need to determine how many divisors 144 has. The formula for finding the number of divisors from the prime factorization \(p_1^{e_1} \times p_2^{e_2} \times \ldots\) is:
\[ (e_1 + 1)(e_2 + 1) \ldots \]
For \(144 = 2^4 \times 3^2\):
- The exponent of 2 is 4, so \(e_1 + 1 = 4 + 1 = 5\).
- The exponent of 3 is 2, so \(e_2 + 1 = 2 + 1 = 3\).

Thus, the total number of divisors is:
\[ 5 \times 3 = 15 \]

### Step 4: Count ordered pairs
Each divisor \(d\) of 144 can form a pair \((d, \frac{144}{d})\). Since the pairs \((m, n)\) and \((n, m)\) are considered different unless \(m = n\), we need to account for this.

### Step 5: Identify the pairs
The divisors of 144 are:
1. \(1\)
2. \(2\)
3. \(3\)
4. \(4\)
5. \(6\)
6. \(8\)
7. \(9\)
8. \(12\)
9. \(16\)
10. \(18\)
11. \(24\)
12. \(36\)
13. \(48\)
14. \(72\)
15. \(144\)

Now, we can pair them:
- \((1, 144)\)
- \((2, 72)\)
- \((3, 48)\)
- \((4, 36)\)
- \((6, 24)\)
- \((8, 18)\)
- \((9, 16)\)
- \((12, 12)\)

### Step 6: Count the unique pairs
For each of the 14 pairs listed above, we can form two ordered pairs except for the pair \((12, 12)\) which only counts once. Therefore:
- Total pairs = \(14 \times 2 + 1 = 29\).

### Step 7: Consider negative pairs
Since both \(m\) and \(n\) can also be negative, we can repeat the same process with negative values:
- Each positive pair \((m, n)\) has a corresponding negative pair \((-m, -n)\).

Thus, we have:
- Total pairs = \(29 + 1\) (for \((12, 12)\)) + \(29\) (for negative pairs) = \(30\).

### Final Answer
The total number of ordered pairs \((m, n)\) that satisfy the equation is:
\[ \boxed{30} \]