The HCF and LCM of two 3 digit numbers are 29 and 4147 respectively. Find the sum of these numbers.

Options

A. 966
B. 696
C. 669
D. 666

Answer

To solve the problem, we need to find two three-digit numbers whose highest common factor (HCF) is 29 and least common multiple (LCM) is 4147. We will follow these steps:

### Step 1: Understand the relationship between HCF, LCM, and the numbers
The relationship between the HCF, LCM, and the two numbers (let's call them \( x \) and \( y \)) can be expressed as:
\[ \text{HCF} \times \text{LCM} = x \times y \]

### Step 2: Substitute the given values
We know that:
- HCF = 29
- LCM = 4147

Substituting these values into the equation gives:
\[ 29 \times 4147 = x \times y \]

### Step 3: Calculate the product of the numbers
Now, we will calculate \( 29 \times 4147 \):
\[ 29 \times 4147 = 120263 \]
So, \( x \times y = 120263 \).

### Step 4: Express the numbers in terms of their coprime factors
Since the HCF is 29, we can express the two numbers as:
\[ x = 29a \quad \text{and} \quad y = 29b \]
where \( a \) and \( b \) are coprime (i.e., their HCF is 1).

### Step 5: Substitute into the product equation
Substituting \( x \) and \( y \) into the product equation gives:
\[ (29a) \times (29b) = 120263 \]
This simplifies to:
\[ 29^2 \times (a \times b) = 120263 \]

### Step 6: Calculate \( a \times b \)
Now, we need to divide both sides by \( 29^2 \):
\[ a \times b = \frac{120263}{29^2} \]
Calculating \( 29^2 \):
\[ 29^2 = 841 \]
Now, calculate \( a \times b \):
\[ a \times b = \frac{120263}{841} = 143 \]

### Step 7: Find pairs of coprime factors of 143
Next, we need to find pairs of coprime integers \( (a, b) \) such that \( a \times b = 143 \). The pairs of factors of 143 are:
- \( (1, 143) \)
- \( (11, 13) \)

Both pairs are coprime.

### Step 8: Calculate the corresponding numbers
Now, we will calculate the corresponding numbers for each pair:
1. For \( (1, 143) \):
- \( x = 29 \times 1 = 29 \) (not a three-digit number)
- \( y = 29 \times 143 = 4147 \) (not a three-digit number)

2. For \( (11, 13) \):
- \( x = 29 \times 11 = 319 \)
- \( y = 29 \times 13 = 377 \)

### Step 9: Calculate the sum of the two numbers
Now, we can find the sum of the two three-digit numbers:
\[ 319 + 377 = 696 \]

### Final Answer
The sum of the two numbers is **696**.

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