The product of two numbers is 20808 and HCF is 17. The number are

Options

A. 136, 153
B. 135, 137
C. 138, 154
D. 136, 154

Answer

To find the two numbers whose product is 20808 and whose HCF is 17, we can follow these steps:

### Step 1: Understand the relationship between the numbers
Given that the HCF (Highest Common Factor) of the two numbers is 17, we can express the two numbers as:
- Let the first number be \( 17A \)
- Let the second number be \( 17B \)

Here, \( A \) and \( B \) are coprime numbers (they have no common factors other than 1).

### Step 2: Set up the equation using the product
The product of the two numbers can be expressed as:
\[ (17A) \times (17B) = 20808 \]
This simplifies to:
\[ 289AB = 20808 \]

### Step 3: Solve for \( AB \)
To find \( AB \), we divide both sides of the equation by 289:
\[ AB = \frac{20808}{289} \]

### Step 4: Perform the division
Now, we need to calculate \( \frac{20808}{289} \). Performing the division gives:
\[ AB = 72 \]

### Step 5: Find coprime pairs that multiply to 72
Next, we need to find pairs of coprime numbers \( A \) and \( B \) such that:
\[ A \times B = 72 \]
The pairs of factors of 72 are:
- (1, 72)
- (2, 36)
- (3, 24)
- (4, 18)
- (6, 12)
- (8, 9)

Among these pairs, the coprime pairs are:
- (1, 72)
- (8, 9)

### Step 6: Calculate the original numbers
Now we can calculate the original numbers using the coprime pairs:
1. For the pair (1, 72):
- First number: \( 17 \times 1 = 17 \)
- Second number: \( 17 \times 72 = 1224 \)

2. For the pair (8, 9):
- First number: \( 17 \times 8 = 136 \)
- Second number: \( 17 \times 9 = 153 \)

### Conclusion
Thus, the two possible pairs of numbers are:
1. \( 17 \) and \( 1224 \)
2. \( 136 \) and \( 153 \)

### Final Answer
The two numbers are \( 136 \) and \( 153 \).

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