Write the smallest number which is divisible by both 306 and 657

Answer

To find the smallest number that is divisible by both 306 and 657, we need to calculate the Least Common Multiple (LCM) of these two numbers. Here is a step-by-step solution:

### Step 1: Prime Factorization
First, we need to find the prime factorization of both numbers.

**For 306:**
- Divide by 2: \( 306 \div 2 = 153 \)
- Divide by 3: \( 153 \div 3 = 51 \)
- Divide by 3 again: \( 51 \div 3 = 17 \)
- 17 is a prime number.

Thus, the prime factorization of 306 is:
\[ 306 = 2^1 \times 3^2 \times 7^1 \]

**For 657:**
- Divide by 3: \( 657 \div 3 = 219 \)
- Divide by 3 again: \( 219 \div 3 = 73 \)
- 73 is a prime number.

Thus, the prime factorization of 657 is:
\[ 657 = 3^2 \times 73^1 \]

### Step 2: Identify the LCM
To find the LCM, we take the highest power of each prime factor from both factorizations.

- For prime number 2: \( 2^1 \) (from 306)
- For prime number 3: \( 3^2 \) (common in both)
- For prime number 7: \( 7^1 \) (from 306)
- For prime number 73: \( 73^1 \) (from 657)

Now, we can write the LCM as:
\[ \text{LCM} = 2^1 \times 3^2 \times 7^1 \times 73^1 \]

### Step 3: Calculate the LCM
Now we will calculate the value of the LCM:
1. Calculate \( 3^2 = 9 \)
2. Multiply \( 2 \times 9 = 18 \)
3. Multiply \( 18 \times 7 = 126 \)
4. Finally, multiply \( 126 \times 73 \)

Calculating \( 126 \times 73 \):
- \( 126 \times 73 = 9186 \)

Thus, the LCM of 306 and 657 is:
\[ \text{LCM} = 9186 \]

### Conclusion
The smallest number that is divisible by both 306 and 657 is:
\[ \boxed{9186} \]

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