Find the least square number which is exactly divisible by 4, 9 and 14.

Options

A. 1008
B. 252
C. 1764
D. 504

Answer

To find the least square number that is exactly divisible by 4, 9, and 14, we can follow these steps:

### Step 1: Find the LCM of the numbers
To begin, we need to find the least common multiple (LCM) of the numbers 4, 9, and 14.

- The prime factorization of the numbers is:
- 4 = 2²
- 9 = 3²
- 14 = 2 × 7

- Now, we take the highest power of each prime factor:
- For 2: the highest power is 2² (from 4)
- For 3: the highest power is 3² (from 9)
- For 7: the highest power is 7¹ (from 14)

- Therefore, the LCM is:
\[ \text{LCM} = 2² \times 3² \times 7 = 4 \times 9 \times 7 \]

### Step 2: Calculate the LCM
Now, we can calculate the LCM:
\[ 4 \times 9 = 36 \]
\[ 36 \times 7 = 252 \]
So, the LCM of 4, 9, and 14 is 252.

### Step 3: Check if the LCM is a perfect square
Next, we need to check if 252 is a perfect square. A perfect square has all prime factors raised to even powers.

- The prime factorization of 252 is:
\[ 252 = 2² \times 3² \times 7¹ \]
- Here, the power of 7 is odd (1), which means 252 is not a perfect square.

### Step 4: Make it a perfect square
To make 252 a perfect square, we need to ensure that all prime factors have even powers. Since the power of 7 is odd, we need to multiply 252 by 7 to make it even.

\[ \text{New number} = 252 \times 7 = 1764 \]

### Step 5: Verify if the new number is a perfect square
Now, let's check if 1764 is a perfect square:
- The prime factorization of 1764 is:
\[ 1764 = 2² \times 3² \times 7² \]
- All prime factors are raised to even powers, confirming that 1764 is a perfect square.

### Conclusion
Thus, the least square number that is exactly divisible by 4, 9, and 14 is **1764**.

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