The sum of three consecutive natural numbers each divisible by 3 is72.What is the largest among them?

Options

A. 21
B. 24
C. 27
D. 36

Answer

To solve the problem step by step, we need to find three consecutive natural numbers that are divisible by 3 and whose sum is 72.

### Step 1: Define the three consecutive natural numbers
Let the three consecutive natural numbers be:
- First number: \( n \)
- Second number: \( n + 1 \)
- Third number: \( n + 2 \)

Since we need these numbers to be divisible by 3, we can express them in terms of a variable that ensures divisibility by 3. Let's denote the first number as \( n = 3k \) for some integer \( k \). Thus, the three consecutive numbers can be expressed as:
- First number: \( 3k \)
- Second number: \( 3k + 1 \)
- Third number: \( 3k + 2 \)

### Step 2: Set up the equation for their sum
The sum of these three numbers can be expressed as:
\[ 3k + (3k + 1) + (3k + 2) = 72 \]
This simplifies to:
\[ 3k + 3k + 1 + 3k + 2 = 72 \]
Combining like terms gives:
\[ 9k + 3 = 72 \]

### Step 3: Solve for \( k \)
Now, we need to isolate \( k \):
\[ 9k + 3 = 72 \]
Subtract 3 from both sides:
\[ 9k = 69 \]
Now, divide both sides by 9:
\[ k = \frac{69}{9} = 7.67 \]
Since \( k \) must be an integer, we need to adjust our approach.

### Step 4: Re-evaluate the consecutive numbers
Instead, let’s denote the three consecutive natural numbers directly as:
- First number: \( 3x \)
- Second number: \( 3x + 3 \)
- Third number: \( 3x + 6 \)

Now, their sum is:
\[ 3x + (3x + 3) + (3x + 6) = 72 \]
This simplifies to:
\[ 9x + 9 = 72 \]

### Step 5: Solve for \( x \)
Now, isolate \( x \):
\[ 9x + 9 = 72 \]
Subtract 9 from both sides:
\[ 9x = 63 \]
Now, divide both sides by 9:
\[ x = 7 \]

### Step 6: Find the three consecutive numbers
Now we can find the three consecutive numbers:
- First number: \( 3x = 3 \times 7 = 21 \)
- Second number: \( 3x + 3 = 21 + 3 = 24 \)
- Third number: \( 3x + 6 = 21 + 6 = 27 \)

### Step 7: Identify the largest number
The largest of these three numbers is:
\[ 27 \]

Thus, the largest among the three consecutive natural numbers is **27**.