Understanding the Problem: Ratio and LCM

We are given three numbers that are in the ratio 5:1:7. We are also given their Least Common Multiple (LCM), which is 7595. The goal is to find the Highest Common Factor (HCF) of these three numbers.

Representing the Numbers

Let the three numbers be represented using the given ratio and a common factor, which is the HCF. Let the HCF be represented by '$x$'.

• The first number = $5x$
• The second number = $1x$ (or simply $x$)
• The third number = $7x$

Calculating LCM using HCF and Ratio

The LCM of numbers represented as $ax$, $bx$, and $cx$ (where $x$ is the HCF) can be calculated as:

LCM = $x \times \text{LCM}(a, b, c)$

In this case, $a=5$, $b=1$, and $c=7$. So, we need to find the LCM of 5, 1, and 7.

Since 5, 1, and 7 are coprime numbers (their only common factor is 1), their LCM is simply their product:

LCM(5, 1, 7) = $5 \times 1 \times 7 = 35$

Now, we can express the LCM of the three numbers (5x, x, 7x) in terms of $x$:

LCM(5x, x, 7x) = $x \times \text{LCM}(5, 1, 7) = x \times 35 = 35x$

Finding the HCF

We are given that the LCM of the three numbers is 7595. We have calculated the LCM as $35x$. Therefore, we can set up the equation:

$$35x = 7595$$

To find the HCF ($x$), we need to solve this equation for $x$:

$$x = \frac{7595}{35}$$

Step-by-Step Calculation

Let's perform the division:

1. Divide 7595 by 35.
2. $7595 \div 35 = 217$

So, the value of $x$ is 217.

Conclusion

Since $x$ represents the Highest Common Factor (HCF) of the three numbers, the HCF is 217.

The three numbers are:

• $5 \times 217 = 1085$
• $1 \times 217 = 217$
• $7 \times 217 = 1519$

The HCF of 1085, 217, and 1519 is indeed 217.