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’s a step-by-step solution:

### Step 1: Prime Factorization of 306
1. Start with 306. Since it is even, divide by 2:
\[ 306 \div 2 = 153 \]
2. Next, factor 153. The sum of the digits (1 + 5 + 3 = 9) is divisible by 3, so divide by 3:
\[ 153 \div 3 = 51 \]
3. Factor 51. The sum of the digits (5 + 1 = 6) is also divisible by 3, so divide by 3 again:
\[ 51 \div 3 = 17 \]
4. Finally, 17 is a prime number.

Thus, the prime factorization of 306 is:
\[ 306 = 2^1 \times 3^2 \times 17^1 \]

### Step 2: Prime Factorization of 657
1. Start with 657. The sum of the digits (6 + 5 + 7 = 18) is divisible by 3, so divide by 3:
\[ 657 \div 3 = 219 \]
2. Factor 219. The sum of the digits (2 + 1 + 9 = 12) is also divisible by 3, so divide by 3 again:
\[ 219 \div 3 = 73 \]
3. Finally, 73 is a prime number.

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

### Step 3: Finding the LCM
To find the LCM, take the highest power of each prime factor from both factorizations:
- From 306: \(2^1\), \(3^2\), \(17^1\)
- From 657: \(3^2\), \(73^1\)

Now, combine these:
\[ \text{LCM} = 2^1 \times 3^2 \times 17^1 \times 73^1 \]

### Step 4: Calculate the LCM
Now we will calculate the LCM:
1. Calculate \(2^1 = 2\)
2. Calculate \(3^2 = 9\)
3. Calculate \(17^1 = 17\)
4. Calculate \(73^1 = 73\)

Now multiply these together:
\[ \text{LCM} = 2 \times 9 \times 17 \times 73 \]

Calculating step-by-step:
1. \(2 \times 9 = 18\)
2. \(18 \times 17 = 306\)
3. \(306 \times 73 = 22338\)

Thus, the smallest number which is divisible by both 306 and 657 is:
\[ \text{LCM} = 22338 \]

### Final Answer
The smallest number which is divisible by both 306 and 657 is **22338**.

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