Finding HCF from Ratio and LCM of Numbers

This problem involves the relationship between the ratio, HCF (Highest Common Factor), and LCM (Least Common Multiple) of three numbers. We are given the ratio of the numbers and their LCM, and we need to find their HCF.

Understanding the Relationship between HCF, Ratio, and Numbers

If three numbers are in the ratio \(a : b : c\), and their HCF is \(H\), then the numbers can be written as \(aH\), \(bH\), and \(cH\). Here, \(a, b, c\) are the terms in the ratio, which are generally taken in their simplest form (i.e., their HCF is 1), but this is not strictly necessary for calculating the LCM of \(aH, bH, cH\). The key relationship is that the LCM of the numbers \(aH, bH, cH\) is equal to \(H \times \text{LCM}(a, b, c)\).

Step-by-Step Solution for HCF Calculation

Let the three numbers be \(N_1\), \(N_2\), and \(N_3\). The given ratio is 3 : 8 : 15. Let the HCF of these three numbers be \(H\). So, the numbers can be represented as:

• \(N_1 = 3H\)
• \(N_2 = 8H\)
• \(N_3 = 15H\)

The LCM of these three numbers is given as 8280.

The formula relating LCM and HCF for numbers in a ratio is: \[ \text{LCM}(N_1, N_2, N_3) = H \times \text{LCM}(\text{ratio terms}) \] In this case, the ratio terms are 3, 8, and 15.

Calculating the LCM of the Ratio Terms

We need to find the LCM of 3, 8, and 15.

• Prime factorization of 3 is \(3^1\).
• Prime factorization of 8 is \(2^3\).
• Prime factorization of 15 is \(3^1 \times 5^1\).

To find the LCM, we take the highest power of each prime factor present:

Setting up the Equation and Solving for HCF

Now we use the formula: \[ \text{LCM}(N_1, N_2, N_3) = H \times \text{LCM}(3, 8, 15) \] We know the LCM of the numbers is 8280 and the LCM of the ratio terms is 120. \[ 8280 = H \times 120 \] To find \(H\), we divide 8280 by 120:

Performing the division:

So, \(H = 69\).

The HCF of the three numbers is 69.

Verification (Optional but Recommended)

The numbers are \(3 \times 69 = 207\), \(8 \times 69 = 552\), and \(15 \times 69 = 1035\).

Let's find the LCM of 207, 552, and 1035.

• \(207 = 3 \times 69 = 3 \times 3 \times 23 = 3^2 \times 23^1\)
• \(552 = 8 \times 69 = 2^3 \times 3^1 \times 23^1\)
• \(1035 = 15 \times 69 = 3 \times 5 \times 3 \times 23 = 3^2 \times 5^1 \times 23^1\)

LCM(207, 552, 1035) = \(2^3 \times 3^2 \times 5^1 \times 23^1 = 8 \times 9 \times 5 \times 23 = 72 \times 5 \times 23 = 360 \times 23 = 8280\). The calculated LCM matches the given LCM, so our HCF value is correct.

Revision Table: HCF and LCM Concepts

Additional Information on HCF and LCM Problems

Problems involving HCF, LCM, and ratios are common in quantitative aptitude sections of various exams. Understanding the fundamental definitions and relationships is crucial.

• When numbers are in a ratio \(a:b:c\), and their HCF is \(H\), the numbers are \(aH, bH, cH\). This is a fundamental way to represent the numbers.
• The relationship \(\text{Product of two numbers} = \text{HCF} \times \text{LCM}\) is only valid for *two* numbers, not three or more in general.
• For three numbers, the relationship between their product, HCF, and LCM is more complex and does not follow a simple multiplicative formula like the one for two numbers.
• However, the relationship \(\text{LCM}(aH, bH, cH) = H \times \text{LCM}(a, b, c)\) used in this problem is always true.

Being able to find the LCM of the ratio terms efficiently (using prime factorization) is key to solving this type of problem quickly.