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