Find the L.C.M. of 198 and 78.

Options

A. L.C.M. = 2572
B. L.C.M. = 2574
C. L.C.M. = 2576
D. L.C.M. = 2578

Answer

To find the L.C.M. (Least Common Multiple) of 198 and 78, we can follow these steps:

### Step 1: Find the prime factorization of each number.

**For 198:**
1. Divide by 2 (the smallest prime number):
\( 198 \div 2 = 99 \)
So, \( 198 = 2 \times 99 \)
2. Next, factor 99:
\( 99 \div 3 = 33 \)
So, \( 99 = 3 \times 33 \)
3. Now factor 33:
\( 33 \div 3 = 11 \)
So, \( 33 = 3 \times 11 \)
4. Therefore, the prime factorization of 198 is:
\( 198 = 2 \times 3^2 \times 11 \)

**For 78:**
1. Divide by 2:
\( 78 \div 2 = 39 \)
So, \( 78 = 2 \times 39 \)
2. Next, factor 39:
\( 39 \div 3 = 13 \)
So, \( 39 = 3 \times 13 \)
3. Therefore, the prime factorization of 78 is:
\( 78 = 2 \times 3 \times 13 \)

### Step 2: Identify the common and unique factors.

- From the factorization:
- For 198: \( 2^1, 3^2, 11^1 \)
- For 78: \( 2^1, 3^1, 13^1 \)

### Step 3: Write down the L.C.M. using the highest powers of all prime factors.

- The L.C.M. will include:
- The highest power of 2: \( 2^1 \)
- The highest power of 3: \( 3^2 \)
- The highest power of 11: \( 11^1 \)
- The highest power of 13: \( 13^1 \)

So, the L.C.M. is:
\[ L.C.M. = 2^1 \times 3^2 \times 11^1 \times 13^1 \]

### Step 4: Calculate the L.C.M.

1. Calculate \( 2 \times 3^2 \):
\( 3^2 = 9 \)
\( 2 \times 9 = 18 \)
2. Now multiply by 11:
\( 18 \times 11 = 198 \)
3. Finally, multiply by 13:
\( 198 \times 13 = 2574 \)

Thus, the L.C.M. of 198 and 78 is **2574**.

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