The HCF and LCM of two numbers are 12 and 420 respectively. Then the number of such pairs is:

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the number of pairs of two numbers whose HCF is 12 and LCM is 420, we can follow these steps:

### Step 1: Understand the relationship between HCF, LCM, and the two numbers

The relationship between the HCF (Highest Common Factor), LCM (Lowest Common Multiple), and the two numbers (let's call them A and B) can be expressed as:
\[ \text{HCF} \times \text{LCM} = A \times B \]

### Step 2: Substitute the given values

We know:
- HCF = 12
- LCM = 420

Substituting these values into the equation gives:
\[ 12 \times 420 = A \times B \]

### Step 3: Calculate the product of A and B

Calculating the left side:
\[ 12 \times 420 = 5040 \]
So, we have:
\[ A \times B = 5040 \]

### Step 4: Express A and B in terms of HCF

Since the HCF of A and B is 12, we can express A and B as:
\[ A = 12a \]
\[ B = 12b \]
where a and b are coprime (their HCF is 1).

### Step 5: Substitute A and B into the product equation

Now substituting A and B into the product equation:
\[ (12a) \times (12b) = 5040 \]
This simplifies to:
\[ 144ab = 5040 \]

### Step 6: Solve for ab

Dividing both sides by 144 gives:
\[ ab = \frac{5040}{144} \]
Calculating the right side:
\[ ab = 35 \]

### Step 7: Find pairs (a, b) such that ab = 35

Now we need to find pairs of (a, b) such that their product is 35 and they are coprime. The factor pairs of 35 are:
- (1, 35)
- (5, 7)

### Step 8: Check if pairs are coprime

Now we check if these pairs are coprime:
- For (1, 35): HCF(1, 35) = 1 (Coprime)
- For (5, 7): HCF(5, 7) = 1 (Coprime)

Both pairs are coprime.

### Step 9: Count the pairs

Since we have two pairs (1, 35) and (5, 7), we can conclude that there are 2 pairs of (a, b).

### Step 10: Calculate the corresponding pairs (A, B)

Now we can find the corresponding pairs (A, B):
1. For (1, 35):
- A = 12 * 1 = 12
- B = 12 * 35 = 420
- Pair: (12, 420)

2. For (5, 7):
- A = 12 * 5 = 60
- B = 12 * 7 = 84
- Pair: (60, 84)

### Final Answer

Thus, the number of such pairs (A, B) is **2**.

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