The product of two 2-digit number is 2160 and their H.C.F is 12. The number are

Options

A. (12, 60)
B. (72,30)
C. (36,60)
D. (60,72)

Answer

To solve the problem, we need to find two 2-digit numbers whose product is 2160 and whose highest common factor (H.C.F) is 12. Let's break this down step by step.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We know the product of the two numbers \( X \) and \( Y \) is \( 2160 \).
- Their H.C.F is \( 12 \).

2. **Expressing the Numbers in Terms of H.C.F**:
- Since the H.C.F of the two numbers is \( 12 \), we can express the numbers as:
\[ X = 12a \quad \text{and} \quad Y = 12b \]
- Here, \( a \) and \( b \) are co-prime numbers (meaning they have no common factors other than 1).

3. **Setting Up the Equation**:
- The product of the two numbers can be expressed as:
\[ X \times Y = (12a) \times (12b) = 144ab \]
- We know that this product equals \( 2160 \):
\[ 144ab = 2160 \]

4. **Solving for \( ab \)**:
- To find \( ab \), we divide both sides by \( 144 \):
\[ ab = \frac{2160}{144} \]
- Performing the division:
\[ ab = 15 \]

5. **Finding Co-prime Pairs**:
- Now we need to find pairs of co-prime numbers \( (a, b) \) such that their product is \( 15 \). The pairs are:
- \( (1, 15) \)
- \( (3, 5) \)

6. **Calculating the Two-Digit Numbers**:
- For each pair, we can calculate \( X \) and \( Y \):
- For \( (1, 15) \):
\[ X = 12 \times 1 = 12 \quad \text{and} \quad Y = 12 \times 15 = 180 \quad \text{(not a 2-digit number)} \]
- For \( (3, 5) \):
\[ X = 12 \times 3 = 36 \quad \text{and} \quad Y = 12 \times 5 = 60 \]

7. **Final Result**:
- The two-digit numbers are \( 36 \) and \( 60 \).

### Conclusion:
The required two-digit numbers are **36 and 60**.