The LCM of two numbers which are in the ratio 3:8 is 144 . Find the two numbers

Answer

To find the two numbers that are in the ratio of 3:8 and have an LCM of 144, we can follow these steps:

### Step 1: Represent the numbers in terms of a variable
Let the two numbers be represented as:
- First number = 3x
- Second number = 8x

### Step 2: Use the formula for LCM
The LCM of two numbers can be calculated using the formula:
\[ \text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)} \]
However, since we are dealing with numbers in a ratio, we can directly calculate the LCM as:
\[ \text{LCM}(3x, 8x) = \frac{3x \times 8x}{\text{GCD}(3x, 8x)} \]

### Step 3: Find GCD of the numbers
The GCD of 3x and 8x is:
\[ \text{GCD}(3x, 8x) = x \]
This is because 3 and 8 have no common factors other than 1.

### Step 4: Substitute GCD into the LCM formula
Now substituting the GCD into the LCM formula:
\[ \text{LCM}(3x, 8x) = \frac{3x \times 8x}{x} \]
This simplifies to:
\[ \text{LCM}(3x, 8x) = 3 \times 8 \times x = 24x \]

### Step 5: Set the LCM equal to the given value
We know from the problem that the LCM is 144:
\[ 24x = 144 \]

### Step 6: Solve for x
To find x, divide both sides by 24:
\[ x = \frac{144}{24} = 6 \]

### Step 7: Find the two numbers
Now we can find the two numbers:
- First number = 3x = 3 × 6 = 18
- Second number = 8x = 8 × 6 = 48

### Final Answer
The two numbers are 18 and 48.

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