Step 1 : Understanding the Question:
This problem explores the relationship between two positive integers and their Greatest Common Divisor (HCF) and Least Common Multiple (LCM). A fundamental theorem in number theory states that for any two numbers, the product of their HCF and LCM is exactly equal to the product of the numbers themselves. Given three out of these four variables (LCM, HCF, and one number), we can use basic algebraic division to find the fourth, missing value. This property is exclusively true for a pair of numbers.
Step 2 : Key Formulas and approach:
The primary relationship used here is:
$$\text{Product of two numbers} = \text{HCF} \times \text{LCM}$$
Let the numbers be $x$ and $y$. Then:
$$x \times y = \text{HCF}(x, y) \times \text{LCM}(x, y)$$
The approach is to plug the known values into this equation and solve for the unknown number.
Step 3 : Detailed Explanation:
Given data: $\text{LCM} = 180$, $\text{HCF} = 6$, and one number ($x$) = 30.
Let the second number be '$y$'.
Apply the formula: $30 \times y = 6 \times 180$.
To find $y$, move 30 to the other side: $y = \frac{6 \times 180}{30}$.
Simplify the division first: $180 \div 30 = 6$.
Now, multiply the remaining numbers: $y = 6 \times 6 = 36$.
Therefore, the second number is 36.
Step 4 : Final Answer:
The other number is 36.