Find the least square number which is divisible by 4, 8, 2, 6 and 12?

Options

A. 36
B. 24
C. 48
D. 144

Answer

To find the least square number that is divisible by 4, 8, 2, 6, and 12, we will follow these steps:

### Step 1: Find the LCM of the numbers
First, we need to find the Least Common Multiple (LCM) of the numbers 4, 8, 2, 6, and 12.

- The prime factorization of each number is:
- 2 = \(2^1\)
- 4 = \(2^2\)
- 8 = \(2^3\)
- 6 = \(2^1 \times 3^1\)
- 12 = \(2^2 \times 3^1\)

### Step 2: Determine the highest powers of each prime factor
Next, we take the highest power of each prime factor from the factorizations:
- For \(2\): The highest power is \(2^3\) (from 8).
- For \(3\): The highest power is \(3^1\) (from 6 and 12).

### Step 3: Calculate the LCM
Now we can calculate the LCM:
\[ \text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24 \]

### Step 4: Ensure the LCM is a perfect square
The LCM we found is 24. However, we need to find the least square number divisible by 24. A perfect square must have even powers of all prime factors.

- The factorization of 24 is \(2^3 \times 3^1\).
- To make the powers even, we need to adjust them:
- For \(2^3\), we need one more \(2\) to make it \(2^4\).
- For \(3^1\), we need one more \(3\) to make it \(3^2\).

### Step 5: Multiply to get the least square number
Now, we multiply the LCM by the necessary factors to make the powers even:
\[ \text{Least square number} = 24 \times 2^1 \times 3^1 = 24 \times 2 \times 3 = 24 \times 6 = 144 \]

### Conclusion
Thus, the least square number that is divisible by 4, 8, 2, 6, and 12 is **144**.

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