The H.C.F. of two numbers , each having three digits, is 17 and their L.C.M. is 714 . The sum of the numbers will be

Options

A. 289
B. 391
C. 221
D. 731

Answer

To solve the problem step by step, we will use the relationship between HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of two numbers.

### Step-by-Step Solution:

1. **Understanding the relationship between HCF and LCM**:
The relationship between the HCF and LCM of two numbers \( a \) and \( b \) is given by:
\[ HCF(a, b) \times LCM(a, b) = a \times b \]
In this case, we know:
\[ HCF = 17 \quad \text{and} \quad LCM = 714 \]

2. **Expressing the numbers in terms of HCF**:
Since the HCF is 17, we can express the two numbers as:
\[ a = 17x \quad \text{and} \quad b = 17y \]
where \( x \) and \( y \) are integers.

3. **Substituting into the LCM formula**:
Now substituting \( a \) and \( b \) into the LCM formula:
\[ LCM(a, b) = \frac{a \times b}{HCF(a, b)} \]
This gives us:
\[ 714 = \frac{(17x) \times (17y)}{17} \]
Simplifying this, we get:
\[ 714 = 17xy \]

4. **Solving for \( xy \)**:
Dividing both sides by 17:
\[ xy = \frac{714}{17} = 42 \]

5. **Finding pairs of factors of 42**:
We need to find pairs of integers \( (x, y) \) such that:
\[ xy = 42 \]
The pairs of factors of 42 are:
- (1, 42)
- (2, 21)
- (3, 14)
- (6, 7)

6. **Choosing valid pairs**:
Since both numbers must be three digits, we need to check which pairs will yield three-digit numbers when multiplied by 17:
- For \( (6, 7) \):
- \( a = 17 \times 6 = 102 \)
- \( b = 17 \times 7 = 119 \)
- Other pairs will yield numbers less than 100.

7. **Calculating the sum of the numbers**:
Now we can find the sum of the two numbers:
\[ 102 + 119 = 221 \]

### Final Answer:
The sum of the two numbers is **221**.

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