The HCF and LCM of two numbers are 11 and 385 respectively. The numbers are

Options

A. 11, 385 or 55, 77
B. 55, 17
C. 11, 385
D. 22, 770

Answer

To find the two numbers given their HCF and LCM, we can use the relationship between HCF, LCM, and the numbers themselves. Here’s a step-by-step solution:

### Step 1: Identify the HCF and LCM
The HCF (Highest Common Factor) is given as 11, and the LCM (Lowest Common Multiple) is given as 385.

### Step 2: Use the relationship between HCF, LCM, and the numbers
The product of the two numbers (let's call them x and y) can be expressed as:
\[ x \times y = \text{HCF} \times \text{LCM} \]
Substituting the given values:
\[ x \times y = 11 \times 385 \]

### Step 3: Calculate the product
Now, calculate the product:
\[ 11 \times 385 = 4235 \]
So, we have:
\[ x \times y = 4235 \]

### Step 4: Express the numbers in terms of HCF
Since 11 is the HCF, we can express the two numbers as:
\[ x = 11a \quad \text{and} \quad y = 11b \]
where a and b are co-prime numbers (they have no common factors other than 1).

### Step 5: Substitute into the product equation
Now substitute x and y into the product equation:
\[ (11a) \times (11b) = 4235 \]
This simplifies to:
\[ 121ab = 4235 \]

### Step 6: Solve for ab
Now divide both sides by 121 to find ab:
\[ ab = \frac{4235}{121} \]
Calculating this gives:
\[ ab = 35 \]

### Step 7: Find pairs of co-prime numbers
Now we need to find pairs of co-prime numbers (a, b) such that their product is 35. The pairs of factors of 35 are:
- (1, 35)
- (5, 7)

### Step 8: Calculate the two numbers
Now, we can calculate the two numbers using the pairs:
1. For the pair (1, 35):
- \( x = 11 \times 1 = 11 \)
- \( y = 11 \times 35 = 385 \)

2. For the pair (5, 7):
- \( x = 11 \times 5 = 55 \)
- \( y = 11 \times 7 = 77 \)

### Conclusion
Thus, the two sets of numbers that satisfy the conditions are:
1. (11, 385)
2. (55, 77)