Solving LCM and HCF Problems: Finding the Other Number

This problem involves the concepts of Least Common Multiple (LCM) and Highest Common Factor (HCF) of two numbers, along with their relationship.

We are given the following information:

• The LCM of two numbers is 22 times their HCF. Mathematically, this can be written as:

\(\text{LCM} = 22 \times \text{HCF}\)

• The sum of their LCM and HCF is 276. Mathematically:

\(\text{LCM} + \text{HCF} = 276\)

• One of the numbers is 132.
• We need to find the other number.

Let's use the given equations to find the values of LCM and HCF.

Substitute the first equation (\(\text{LCM} = 22 \times \text{HCF}\)) into the second equation (\(\text{LCM} + \text{HCF} = 276\)):

\((22 \times \text{HCF}) + \text{HCF} = 276\)

\(23 \times \text{HCF} = 276\)

Now, solve for HCF:

\(\text{HCF} = \frac{276}{23}\)

Performing the division:

\(\text{HCF} = 12\)

Now that we have the HCF, we can find the LCM using the relation \(\text{LCM} = 22 \times \text{HCF}\):

\(\text{LCM} = 22 \times 12\)

Performing the multiplication:

\(\text{LCM} = 264\)

We can quickly check if the sum is 276: \(264 + 12 = 276\). This matches the given information, so our calculated LCM and HCF values are correct.

Now, we need to find the other number. A fundamental property relating LCM and HCF of two positive integers is that the product of the two numbers is equal to the product of their LCM and HCF.

\(\text{Product of two numbers} = \text{LCM} \times \text{HCF}\)

Let the two numbers be \(\text{Number}_1\) and \(\text{Number}_2\). We are given \(\text{Number}_1 = 132\).

\(\text{Number}_1 \times \text{Number}_2 = \text{LCM} \times \text{HCF}\)

Substitute the known values:

\(132 \times \text{Number}_2 = 264 \times 12\)

Now, solve for \(\text{Number}_2\):

\(\text{Number}_2 = \frac{264 \times 12}{132}\)

We can simplify this expression. Notice that 264 is exactly twice 132 (\(132 \times 2 = 264\)).

\(\text{Number}_2 = \frac{(2 \times 132) \times 12}{132}\)

Cancel out 132 from the numerator and denominator:

\(\text{Number}_2 = 2 \times 12\)

\(\text{Number}_2 = 24\)

Thus, the other number is 24.

Let's verify this: The two numbers are 132 and 24. Their HCF is 12 (since \(132 = 12 \times 11\) and \(24 = 12 \times 2\), and 11 and 2 are coprime). Their LCM is \(12 \times 11 \times 2 = 12 \times 22 = 264\). LCM (264) is 22 times HCF (12), and LCM + HCF (\(264 + 12 = 276\)). The calculations match the given conditions.

Revision Table: LCM and HCF Properties

Additional Information: Finding LCM and HCF

You can find the LCM and HCF of numbers using the prime factorization method. Here’s how for 132 and 24:

• Prime Factorization:
• \(132 = 2 \times 66 = 2 \times 2 \times 33 = 2^2 \times 3^1 \times 11^1\)
• \(24 = 2 \times 12 = 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1\)
• HCF: Take the lowest power of common prime factors. Common factors are 2 and 3. Lowest power of 2 is \(2^2\). Lowest power of 3 is \(3^1\). HCF = \(2^2 \times 3^1 = 4 \times 3 = 12\).
• LCM: Take the highest power of all prime factors involved. Highest power of 2 is \(2^3\). Highest power of 3 is \(3^1\). Highest power of 11 is \(11^1\). LCM = \(2^3 \times 3^1 \times 11^1 = 8 \times 3 \times 11 = 24 \times 11 = 264\).

These values match the ones we found using the problem's conditions, confirming our answer for the other number.