Understanding the Problem: Ratio, LCM, and HCF

This problem asks us to find the Highest Common Factor (HCF) of three numbers when we know their ratio and their Least Common Multiple (LCM).

• The ratio of the three numbers is given as $3:4:5$.
• The L.C.M. of these three numbers is $2400$.

Let the three numbers be represented by $3x$, $4x$, and $5x$. Here, '$x$' represents the H.C.F. of these three numbers. This is because when we divide each number by their H.C.F., we get the simplest form of the ratio, which is $3:4:5$.

Calculating the LCM using HCF and Ratio

There's a useful relationship between the H.C.F., L.C.M., and the numbers themselves. For numbers $a, b, c$ with H.C.F. $= x$ and ratio $p:q:r$, we can say:

The numbers are $px, qx, rx$.

Their L.C.M. can be calculated as: L.C.M. $= x \times \text{L.C.M.}(p, q, r)$

In our case, $p=3$, $q=4$, and $r=5$. So, the numbers are $3x$, $4x$, and $5x$.

First, let's find the L.C.M. of the ratio parts, which are $3$, $4$, and $5$. Since $3$, $4$, and $5$ do not share any common factors other than $1$ (they are coprime relative to each other in this context), their L.C.M. is simply their product:

L.C.M.$(3, 4, 5) = 3 \times 4 \times 5 = 60$.

Now, we can use the formula for the L.C.M. of the three numbers:

L.C.M. of the numbers $= x \times \text{L.C.M.}(3, 4, 5)$

L.C.M. of the numbers $= x \times 60 = 60x$.

Solving for the HCF

We are given that the L.C.M. of the numbers is $2400$. We can set up an equation:

To find the value of $x$ (which is the H.C.F.), we need to solve this equation:

$x = \frac{2400}{60}$

Dividing $2400$ by $60$:

$x = 40$

So, the H.C.F. of the three numbers is $40$.

Finding the Numbers (Optional Verification)

We can also find the actual numbers using the H.C.F. we just found:

• First number $= 3x = 3 \times 40 = 120$
• Second number $= 4x = 4 \times 40 = 160$
• Third number $= 5x = 5 \times 40 = 200$

Let's quickly verify if the L.C.M. of $120$, $160$, and $200$ is indeed $2400$.

• Prime factorization of $120 = 2^3 \times 3 \times 5$
• Prime factorization of $160 = 2^5 \times 5$
• Prime factorization of $200 = 2^3 \times 5^2$

The L.C.M. is found by taking the highest power of each prime factor present:

L.C.M. $= 2^5 \times 3^1 \times 5^2 = 32 \times 3 \times 25 = 96 \times 25 = 2400$.

The calculation matches the given L.C.M., confirming our H.C.F. is correct.

Conclusion

The H.C.F. of the three numbers is $40$.