Three numbers are in the ratio 2 : 3 : 4. If their LCM is 240, the smaller of the three numbers is:

Options

A. 40
B. 60
C. 20
D. 80

Answer

To solve the problem step by step, we will follow the given information and apply the necessary mathematical concepts.

### Step 1: Understand the Ratio
The three numbers are in the ratio 2:3:4. We can represent these numbers as:
- \( N_1 = 2k \)
- \( N_2 = 3k \)
- \( N_3 = 4k \)

### Step 2: Use the LCM Information
We know that the LCM of these three numbers is 240. We need to find the LCM of \( 2k, 3k, \) and \( 4k \).

### Step 3: Calculate the LCM
The LCM of \( 2k, 3k, \) and \( 4k \) can be calculated as follows:
- First, we can factor out \( k \):
\[ \text{LCM}(2k, 3k, 4k) = k \cdot \text{LCM}(2, 3, 4) \]
- Now, we find the LCM of the coefficients \( 2, 3, \) and \( 4 \):
- The prime factorization gives us:
- \( 2 = 2^1 \)
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)
- The LCM takes the highest power of each prime:
- \( 2^2 \) from \( 4 \)
- \( 3^1 \) from \( 3 \)
- Thus,
\[ \text{LCM}(2, 3, 4) = 2^2 \cdot 3^1 = 4 \cdot 3 = 12 \]

### Step 4: Set Up the Equation
Now we substitute back into the LCM equation:
\[ \text{LCM}(2k, 3k, 4k) = k \cdot 12 \]
Given that this LCM equals 240, we have:
\[ k \cdot 12 = 240 \]

### Step 5: Solve for \( k \)
To find \( k \), we divide both sides by 12:
\[ k = \frac{240}{12} = 20 \]

### Step 6: Calculate the Numbers
Now that we have \( k \), we can find the three numbers:
- \( N_1 = 2k = 2 \cdot 20 = 40 \)
- \( N_2 = 3k = 3 \cdot 20 = 60 \)
- \( N_3 = 4k = 4 \cdot 20 = 80 \)

### Step 7: Identify the Smaller Number
From the values calculated:
- \( N_1 = 40 \)
- \( N_2 = 60 \)
- \( N_3 = 80 \)

The smallest of these numbers is \( N_1 = 40 \).

### Final Answer
The smaller of the three numbers is **40**.

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