What is the least common multiple of 30, 20 and 70?

Options

A. 40
B. 42
C. 120
D. 420

Answer

To find the least common multiple (LCM) of the numbers 30, 20, and 70, we can follow these steps:

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

- **30** can be factored into primes as:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

- **20** can be factored into primes as:
\[ 20 = 2^2 \times 5^1 \]

- **70** can be factored into primes as:
\[ 70 = 2^1 \times 5^1 \times 7^1 \]

### Step 2: Identify the Highest Powers of Each Prime
Next, we take the highest power of each prime number that appears in the factorizations.

- For the prime **2**, the highest power is \(2^2\) (from 20).
- For the prime **3**, the highest power is \(3^1\) (from 30).
- For the prime **5**, the highest power is \(5^1\) (common in all three).
- For the prime **7**, the highest power is \(7^1\) (from 70).

### Step 3: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:
\[ \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1 \]

Calculating this step by step:
1. Calculate \(2^2 = 4\)
2. Multiply by \(3^1\):
\[ 4 \times 3 = 12 \]
3. Multiply by \(5^1\):
\[ 12 \times 5 = 60 \]
4. Finally, multiply by \(7^1\):
\[ 60 \times 7 = 420 \]

Thus, the least common multiple (LCM) of 30, 20, and 70 is:
\[ \text{LCM} = 420 \]

### Final Answer
The least common multiple of 30, 20, and 70 is **420**.

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