Find two 3-digit numbers whose L.C.M is 6188 & H.C.F is 68

Options

A. 680 & 328
B. 612 & 748
C. 884 & 476
D. 476 & 748

Answer

To find two 3-digit numbers whose L.C.M is 6188 and H.C.F is 68, we can follow these steps:

### Step 1: Understand the relationship between LCM, HCF, and the numbers
The relationship between two numbers \( a \) and \( b \) is given by the formula:
\[ a \times b = \text{L.C.M} \times \text{H.C.F} \]
Here, we know that:
- L.C.M = 6188
- H.C.F = 68

### Step 2: Express the numbers in terms of HCF
Let the two numbers be \( a = 68x \) and \( b = 68y \), where \( x \) and \( y \) are co-prime integers.

### Step 3: Substitute into the relationship
Substituting \( a \) and \( b \) into the formula gives:
\[ (68x) \times (68y) = 6188 \times 68 \]
This simplifies to:
\[ 68^2 \times (x \times y) = 6188 \times 68 \]

### Step 4: Simplify the equation
Dividing both sides by \( 68 \):
\[ 68 \times (x \times y) = 6188 \]
Now, divide both sides by 68:
\[ x \times y = \frac{6188}{68} = 91 \]

### Step 5: Find co-prime pairs of factors of 91
The pairs of co-prime factors of 91 are:
- \( (1, 91) \)
- \( (7, 13) \)

### Step 6: Choose the appropriate pair
Since we need both numbers to be 3-digit numbers, we will use the pair \( (7, 13) \).

### Step 7: Calculate the numbers
Now, substituting back to find the numbers:
- For \( x = 7 \):
\[ a = 68 \times 7 = 476 \]
- For \( y = 13 \):
\[ b = 68 \times 13 = 884 \]

### Step 8: Final answer
Thus, the two 3-digit numbers are **476** and **884**.

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