Q 26. The HCF of 2 numbers is 37 and their sum is 518. What is the difference between numbers, if the difference between them is the minimum possible?

1. 148 
2. 111 
3. 74 
4. 37 
5. None of these

Let's solve this problem step by step:

### Given:
- The HCF (Highest Common Factor) of two numbers is **37**.
- The sum of the two numbers is **518**.
- We need to find the minimum possible difference between the two numbers.

### Let the two numbers be \( a \) and \( b \).

#### Step 1: Express the numbers using their HCF
Since the HCF of the two numbers is **37**, the two numbers can be written as:
\[
a = 37x \quad \text{and} \quad b = 37y
\]
where \( x \) and \( y \) are coprime integers (i.e., their HCF is 1).

#### Step 2: Use the given sum
The sum of the two numbers is given as:
\[
a + b = 518
\]
Substituting the expressions for \( a \) and \( b \):
\[
37x + 37y = 518
\]
Dividing through by 37:
\[
x + y = \frac{518}{37} = 14
\]
So, \( x + y = 14 \).

#### Step 3: Minimize the difference
To minimize the difference \( |a - b| \), we want to minimize \( |x - y| \). Since \( x + y = 14 \), the minimum possible difference occurs when \( x \) and \( y \) are as close as possible.

The closest integer pair for \( x \) and \( y \) that satisfies \( x + y = 14 \) is:
\[
x = 7 \quad \text{and} \quad y = 7
\]

#### Step 4: Calculate the difference
Now, let's calculate the difference between the two numbers:
\[
a - b = 37x - 37y = 37(x - y)
\]
Substituting \( x = 8 \) and \( y = 6
")

 Then, the enrst

Q 26. The HCF of 2 numbers is 37 and their sum is 518. What is the difference between numbers, if the difference between them is the minimum possible?
1. 148 
2. 111 
3. 74 
4. 37 
5. None of these

Solve a number theory problem involving the HCF of two numbers (37) and their sum (518). Learn how to find the minimum possible difference between the numbers using concepts like HCF, coprime integers, and linear equations. Suitable for high school students in Grades 9-12.

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