Find the least square number which is exactly divisible by 3, 4, 5, 6 and 8.

Options

A. 360
B. 240
C. 160
D. 120

Answer

To find the least square number that is exactly divisible by 3, 4, 5, 6, and 8, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Numbers**: The numbers we need to consider are 3, 4, 5, 6, and 8.

2. **Prime Factorization**:
- 3 is a prime number: \(3 = 3^1\)
- 4 can be factored into primes: \(4 = 2^2\)
- 5 is a prime number: \(5 = 5^1\)
- 6 can be factored into primes: \(6 = 2^1 \times 3^1\)
- 8 can be factored into primes: \(8 = 2^3\)

3. **List the Prime Factors**:
- From the factorizations, we have:
- \(2\) appears in \(4\) (as \(2^2\)), \(6\) (as \(2^1\)), and \(8\) (as \(2^3\)).
- \(3\) appears in \(3\) (as \(3^1\)) and \(6\) (as \(3^1\)).
- \(5\) appears in \(5\) (as \(5^1\)).

4. **Determine the LCM**:
- The LCM is found by taking the highest power of each prime factor:
- For \(2\): the highest power is \(2^3\) (from 8).
- For \(3\): the highest power is \(3^1\) (from 3 and 6).
- For \(5\): the highest power is \(5^1\) (from 5).
- Therefore, the LCM is:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]

5. **Calculate the LCM**:
- First, calculate \(8 \times 3 = 24\).
- Then, calculate \(24 \times 5 = 120\).
- So, the LCM of 3, 4, 5, 6, and 8 is \(120\).

6. **Adjust for Square Number**:
- A square number must have even powers for all prime factors.
- The LCM \(120\) can be expressed in prime factorization as:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]
- To convert this into a square number, we need to make the powers even:
- For \(2^3\), the next even power is \(2^4\).
- For \(3^1\), the next even power is \(3^2\).
- For \(5^1\), the next even power is \(5^2\).

7. **Form the Square Number**:
- The least square number is:
\[ \text{Least Square Number} = 2^4 \times 3^2 \times 5^2 \]

8. **Calculate the Least Square Number**:
- Calculate \(2^4 = 16\), \(3^2 = 9\), and \(5^2 = 25\).
- Now, multiply these together:
\[ 16 \times 9 = 144 \]
\[ 144 \times 25 = 3600 \]

9. **Final Answer**:
- The least square number which is exactly divisible by 3, 4, 5, 6, and 8 is \(3600\).