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 the two numbers. Here’s a step-by-step solution:

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

- For **20**:
- 20 can be factored as \( 2 \times 10 \)
- 10 can be factored as \( 2 \times 5 \)
- Therefore, the prime factorization of 20 is:
\[ 20 = 2^2 \times 5^1 \]

- For **24**:
- 24 can be factored as \( 2 \times 12 \)
- 12 can be factored as \( 2 \times 6 \)
- 6 can be factored as \( 2 \times 3 \)
- Therefore, the prime factorization of 24 is:
\[ 24 = 2^3 \times 3^1 \]

### Step 2: Determine the LCM
To find the LCM, we take the highest power of each prime factor from the factorizations.

- For the prime factor **2**:
- The highest power between \( 2^2 \) (from 20) and \( 2^3 \) (from 24) is \( 2^3 \).

- For the prime factor **3**:
- The highest power is \( 3^1 \) (from 24).

- For the prime factor **5**:
- The highest power is \( 5^1 \) (from 20).

Now, we can write the LCM as:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 \]

### Step 3: Calculate the LCM
Now we calculate the LCM:
\[ 2^3 = 8 \]
\[ 3^1 = 3 \]
\[ 5^1 = 5 \]
Now, multiply these together:
\[ \text{LCM} = 8 \times 3 \times 5 \]

Calculating step-by-step:
1. \( 8 \times 3 = 24 \)
2. \( 24 \times 5 = 120 \)

Thus, the LCM of 20 and 24 is:
\[ \text{LCM} = 120 \]

### Conclusion
The least positive integer which is divisible by both 20 and 24 is **120**.

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