Find the least perfect square which is completely divisible by 8, 9 and 12.

Options

A. 72
B. 144
C. 288
D. 36

Answer

To find the least perfect square that is completely divisible by 8, 9, and 12, we will follow these steps:

### Step 1: Find the LCM of 8, 9, and 12
To find the least common multiple (LCM), we first find the prime factorization of each number:
- 8 = 2^3
- 9 = 3^2
- 12 = 2^2 × 3^1

Next, we take the highest power of each prime factor:
- For 2: the highest power is 2^3 (from 8)
- For 3: the highest power is 3^2 (from 9)

Now, we can calculate the LCM:
LCM = 2^3 × 3^2 = 8 × 9 = 72

### Step 2: Check if the LCM is a perfect square
The LCM we found is 72. We need to check if 72 is a perfect square. A perfect square has all prime factors raised to even powers.

- The prime factorization of 72 is:
- 2^3 (odd power)
- 3^2 (even power)

Since 2 is raised to an odd power, 72 is not a perfect square.

### Step 3: Find the smallest k such that 72k is a perfect square
We need to find a value of k such that 72k is a perfect square. Since 72 = 2^3 × 3^2, we need to make the power of 2 even.

To make 2^3 even, we can multiply by 2^1 (to make it 2^4):
- Therefore, we can set k = 2 (which gives us 2^1).

Now we can calculate:
k = 2
72k = 72 × 2 = 144

### Step 4: Verify if 144 is a perfect square
Now we check if 144 is a perfect square:
- The prime factorization of 144 is:
- 144 = 12^2 = (2^2 × 3^1)^2 = 2^4 × 3^2

Both prime factors have even powers, confirming that 144 is a perfect square.

### Conclusion
The least perfect square that is completely divisible by 8, 9, and 12 is **144**.

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