The HCF of two numbers is 12 and their product is 31104. How many pairs of such numbers are possible ?

Answer

To solve the problem, we need to determine how many pairs of numbers can be formed given the conditions that their highest common factor (HCF) is 12 and their product is 31104.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
Since the HCF of the two numbers is 12, we can express the two numbers as:
\[ a = 12x \quad \text{and} \quad b = 12y \]
where \(x\) and \(y\) are coprime numbers (i.e., their HCF is 1).

2. **Using the Product**:
The product of the two numbers can be expressed as:
\[ a \cdot b = (12x)(12y) = 144xy \]
We know from the problem statement that:
\[ a \cdot b = 31104 \]
Therefore, we can set up the equation:
\[ 144xy = 31104 \]

3. **Solving for \(xy\)**:
To find \(xy\), we divide both sides of the equation by 144:
\[ xy = \frac{31104}{144} \]
Now we perform the division:
\[ xy = 216 \]

4. **Finding Coprime Pairs**:
Next, we need to find pairs of coprime factors of 216. We start by determining the prime factorization of 216:
\[ 216 = 2^3 \times 3^3 \]

5. **Finding Factor Pairs**:
We need to find pairs \((x, y)\) such that \(xy = 216\) and \(x\) and \(y\) are coprime. The factors of 216 are:
- 1 and 216
- 2 and 108
- 3 and 72
- 4 and 54
- 6 and 36
- 8 and 27
- 9 and 24
- 12 and 18

Now we check which of these pairs are coprime:
- (1, 216) are coprime.
- (2, 108) are not coprime.
- (3, 72) are not coprime.
- (4, 54) are not coprime.
- (6, 36) are not coprime.
- (8, 27) are coprime.
- (9, 24) are not coprime.
- (12, 18) are not coprime.

The coprime pairs we found are:
- (1, 216)
- (8, 27)

6. **Counting the Pairs**:
We have found two pairs of coprime factors of 216. Therefore, the pairs of numbers \( (a, b) \) that satisfy the conditions of the problem are:
- For \( (1, 216) \): \( (12 \times 1, 12 \times 216) = (12, 2592) \)
- For \( (8, 27) \): \( (12 \times 8, 12 \times 27) = (96, 324) \)

Hence, the total number of pairs of such numbers is **2**.

### Final Answer:
The number of pairs of such numbers is **2**.