Calculate the least positive integer which is divisible by 20 and 24.

Options

A. 120
B. 200
C. 150
D. 480

Answer

To find the least positive integer that is divisible by both 20 and 24, we need to calculate the Least Common Multiple (LCM) of these two numbers. Here’s how to do it step by step:

### Step 1: Prime Factorization
First, we will find the prime factorization of both numbers.

- **For 20**:
- 20 = 2 × 10
- 10 = 2 × 5
- So, the prime factorization of 20 is: \( 20 = 2^2 \times 5^1 \)

- **For 24**:
- 24 = 2 × 12
- 12 = 2 × 6
- 6 = 2 × 3
- So, the prime factorization of 24 is: \( 24 = 2^3 \times 3^1 \)

### Step 2: Identify the Highest Powers
Next, we identify the highest powers of all prime factors involved in the factorizations.

- For the prime number **2**, the highest power is \( 2^3 \) (from 24).
- For the prime number **3**, the highest power is \( 3^1 \) (from 24).
- For the prime number **5**, the highest power is \( 5^1 \) (from 20).

### Step 3: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:

\[ LCM = 2^3 \times 3^1 \times 5^1 \]

Calculating this step by step:

1. Calculate \( 2^3 = 8 \)
2. Calculate \( 3^1 = 3 \)
3. Calculate \( 5^1 = 5 \)

Now multiply these results together:

\[ LCM = 8 \times 3 \times 5 \]

Calculating this:

1. First, \( 8 \times 3 = 24 \)
2. Then, \( 24 \times 5 = 120 \)

Thus, the least positive integer which is divisible by both 20 and 24 is **120**.

### Final Answer
The least positive integer that is divisible by 20 and 24 is **120**.

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