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The L.C.M. of two numbers is 120 and their [tex]\(\text{G.C.F.}\)[/tex] is 6. If one of the numbers is 24, find the other number.

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To find the other number when the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) of two numbers are given, along with one of the numbers, you can use the relationship between LCM and GCD.

The relationship between the two numbers (let's call them [tex]\(a\)[/tex] and [tex]\(b\)[/tex]), their LCM, and their GCD is given by the formula:

[tex]\[ \text{LCM}(a, b) \times \text{GCD}(a, b) = a \times b \][/tex]

Here are the steps to solve the problem:

1. Understand the Given Values:
- LCM of the two numbers is 120.
- GCD of the two numbers is 6.
- One of the numbers, [tex]\(a\)[/tex], is 24.

2. Set up the Equation:
Using the relationship above:
[tex]\[ 120 \times 6 = 24 \times b \][/tex]

3. Solve for the Unknown Number ([tex]\(b\)[/tex]):
To find [tex]\(b\)[/tex], we rearrange the equation:
[tex]\[ b = \frac{120 \times 6}{24} \][/tex]

4. Calculate the Value:
[tex]\[ b = \frac{720}{24} \][/tex]
[tex]\[ b = 30 \][/tex]

Therefore, the other number is 30.
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