Question

Two numbers are in the ratio 3 : 5 and their LCM is 180. Find the HCF of these two numbers.

Hint:

Alternatively, use the formula:
\[ \text{Product of numbers} = \text{HCF} \times \text{LCM} \]
\[ (3x) \times (5x) = x \times 180 \implies 15x^2 = 180x \implies 15x = 180 \implies x = 12 \]

Answer 1

Step 1: Understanding the Question:
We are given the ratio of two numbers and their Least Common Multiple (LCM). We need to determine the Highest Common Factor (HCF) of these two numbers.

Step 2: Key Formula or Approach:
Let the two numbers be \(3x\) and \(5x\), where \(x\) is their HCF.
The LCM of \(3x\) and \(5x\) is given by:
\[ \text{LCM}(3x, 5x) = 3 \times 5 \times x = 15x \]

Step 3: Detailed Explanation:

• Express the numbers in terms of a common factor \(x\):
Let the numbers be \(a = 3x\) and \(b = 5x\).
Here, \(x\) represents the Highest Common Factor (HCF) because 3 and 5 are co-prime.

• Formulate the equation using LCM:
\[ \text{LCM} = 15x \]
Given that LCM = 180:
\[ 15x = 180 \]

• Solve for \(x\):
\[ x = \frac{180}{15} = 12 \]

• Since \(x\) is the HCF, we have:
\[ \text{HCF} = 12 \]


Step 4: Final Answer:
The HCF of the two numbers is 12.