The HCF and LCM of two numbers are 99 and 765, respectively. If one of the numbers is 45, what is the other number?

Step 1: Recall the relationship between HCF, LCM, and two numbers

For any two positive integers, the product of the numbers is always equal to the product of their Highest Common Factor (HCF) and Least Common Multiple (LCM). This is a fundamental property used to find an unknown number when the HCF, LCM, and one number are given.

Step 2: Substitute the given values into the formula

We are given the HCF as 99, the LCM as 765, and one of the numbers as 45. Let the other number be represented by 'Other Number'. We substitute these values into the formula from the previous step.

Step 3: Calculate the product of HCF and LCM

First, we calculate the product of the HCF and LCM. Multiplying 99 by 765 gives us 75735. This product represents the product of the two numbers.

Step 4: Solve for the other number

Now we have the equation $45 \times \text{Other Number} = 75735$. To find the 'Other Number', we need to divide the product by the known number, 45.

Step 5: Perform the division

Dividing 75735 by 45 yields 1683. Therefore, the other number is 1683.

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