Two numbers have an HCF of 18 and an LCM of 1512. If one of the numbers is 126, find the positive difference between these number.

The correct option is 90.

To find the positive difference between the two numbers, we can use a fundamental property that relates the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two numbers.

Fundamental Property:
For any two positive numbers, the product of the numbers is equal to the product of their HCF and LCM.

$\text{First Number}\times \text{Second Number}=\text{HCF}\times \text{LCM}$

Step 1: Identify the given values
Let the first number be $a=126$ .
Let the second number be $b$ .
Given $\text{HCF}=18$ and $\text{LCM}=1512$ .

Step 2: Find the second number (b)
Substitute the given values into the formula:

$126\times b=18\times 1512$

Now, solve for $b$ :

$b=\frac{18\times 1512}{126}$

Simplifying the fraction (since $126÷18=7$ ):

$b=\frac{1512}{7}$

Performing the division:

$b=216$

So, the two numbers are 126 and 216.

Step 3: Find the positive difference between the two numbers
The positive difference is given by subtracting the smaller number from the larger number:

$\text{Difference}=216-126=90$

Hence, the positive difference between these two numbers is 90.