The LCM of 60, 90 and 180 is:

Options

A. 720
B. 360
C. 180
D. 90

Answer

To find the LCM (Least Common Multiple) of the numbers 60, 90, and 180, we will follow these steps:

### Step 1: Prime Factorization of Each Number
First, we need to find the prime factorization of each of the three numbers.

- **For 60**:
- 60 can be factored as \( 60 = 2 \times 30 \)
- \( 30 = 2 \times 15 \)
- \( 15 = 3 \times 5 \)
- Therefore, the prime factorization of 60 is:
\[ 60 = 2^2 \times 3^1 \times 5^1 \]

- **For 90**:
- 90 can be factored as \( 90 = 9 \times 10 \)
- \( 9 = 3^2 \)
- \( 10 = 2 \times 5 \)
- Therefore, the prime factorization of 90 is:
\[ 90 = 2^1 \times 3^2 \times 5^1 \]

- **For 180**:
- 180 can be factored as \( 180 = 2 \times 90 \)
- We already know the factorization of 90, so:
\[ 180 = 2^1 \times (2^1 \times 3^2 \times 5^1) = 2^2 \times 3^2 \times 5^1 \]

### Step 2: Identify the Highest Power of Each Prime Factor
Next, we identify the highest power of each prime factor from the factorizations:

- For \( 2 \):
- Highest power is \( 2^2 \) (from 60 and 180)

- For \( 3 \):
- Highest power is \( 3^2 \) (from 90 and 180)

- For \( 5 \):
- Highest power is \( 5^1 \) (from all three numbers)

### Step 3: Calculate the LCM
Now we can calculate the LCM by multiplying the highest powers of all prime factors together:

\[ \text{LCM} = 2^2 \times 3^2 \times 5^1 \]

Calculating this step-by-step:

1. Calculate \( 2^2 = 4 \)
2. Calculate \( 3^2 = 9 \)
3. Multiply \( 4 \times 9 = 36 \)
4. Finally, multiply \( 36 \times 5 = 180 \)

### Conclusion
Thus, the LCM of 60, 90, and 180 is:

\[ \text{LCM} = 180 \]

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