Finding the Other Number Using HCF and LCM

This problem requires us to find one of two numbers when we know their Highest Common Factor (HCF), their Least Common Multiple (LCM), and the value of the other number. We can solve this using a fundamental property that connects these four values.

Key Formula for HCF and LCM Problems

For any two positive integers, say '$a$' and '$b$', the product of the numbers is equal to the product of their HCF and LCM.

The formula is:

$ a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b) $

Step-by-Step Calculation

Let the two numbers be '$a$' and '$b$'. We are given:

• HCF = 11
• LCM = 330
• One number (let's say '$a$') = 110
• We need to find the other number ('$b$').

Using the formula:

$ a \times b = \text{HCF} \times \text{LCM} $

Substitute the known values into the equation:

$ 110 \times b = 11 \times 330 $

Now, we need to solve for '$b$'. First, calculate the product on the right side:

$ 11 \times 330 = 3630 $

So the equation becomes:

$ 110 \times b = 3630 $

To find '$b$', divide both sides by 110:

$ b = \frac{3630}{110} $

We can simplify this division. Notice that both numbers end in zero, so we can divide both by 10:

$ b = \frac{363}{11} $

Now perform the division:

$ b = 33 $

Therefore, the other number is 33.