The LCM of 76 and 57 is :
A)114
B)171
C)152
D)228

Options

A. 114
B. 171
C. 152
D. 228

Answer

To find the Least Common Multiple (LCM) of 76 and 57, we can follow these steps:

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

- **For 76**:
- 76 can be divided by 2 (the smallest prime number):
\[ 76 \div 2 = 38 \]
- 38 can also be divided by 2:
\[ 38 \div 2 = 19 \]
- 19 is a prime number.

So, the prime factorization of 76 is:
\[ 76 = 2^2 \times 19^1 \]

- **For 57**:
- 57 can be divided by 3 (the smallest prime number that divides it):
\[ 57 \div 3 = 19 \]
- 19 is a prime number.

So, the prime factorization of 57 is:
\[ 57 = 3^1 \times 19^1 \]

### Step 2: Identify the Highest Powers of Each Prime Factor
Next, we identify the highest power of each prime factor present in both factorizations.

- The prime factors we have are 2, 3, and 19.
- From 76, we have:
- \(2^2\)
- \(19^1\)
- From 57, we have:
- \(3^1\)
- \(19^1\)

Now we take the highest power of each prime:
- For 2: \(2^2\) (from 76)
- For 3: \(3^1\) (from 57)
- For 19: \(19^1\) (common in both)

### Step 3: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:
\[ \text{LCM} = 2^2 \times 3^1 \times 19^1 \]
Calculating this step-by-step:
1. Calculate \(2^2 = 4\)
2. Multiply by \(3\):
\[ 4 \times 3 = 12 \]
3. Finally, multiply by \(19\):
\[ 12 \times 19 = 228 \]

### Final Answer
Thus, the LCM of 76 and 57 is:
\[ \text{LCM} = 228 \]

The correct option is **D) 228**.

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