For what value of N, 270N will be perfect square, where 270N is a 4-digit number ?

Options

A. 1
B. 6
C. 4
D. 9

Answer

To find the value of \( N \) such that \( 270N \) is a perfect square and a four-digit number, we can follow these steps:

### Step 1: Determine the range of \( N \)
Since \( 270N \) is a four-digit number, we need to find the range for \( N \).

The smallest four-digit number is 1000, and the largest is 9999. Therefore, we can set up the following inequalities:

\[ 1000 \leq 270N \leq 9999 \]

### Step 2: Solve for \( N \)
Now, we divide the entire inequality by 270 to isolate \( N \):

\[ \frac{1000}{270} \leq N \leq \frac{9999}{270} \]

Calculating the values:

\[ \frac{1000}{270} \approx 3.70 \quad \text{and} \quad \frac{9999}{270} \approx 37.04 \]

Thus, the integer values of \( N \) can range from 4 to 37.

### Step 3: Check for perfect squares
Next, we need to check which values of \( N \) make \( 270N \) a perfect square.

We can express \( 270 \) in its prime factorization:

\[ 270 = 2 \times 3^3 \times 5 \]

For \( 270N \) to be a perfect square, all prime factors must have even exponents. The current exponents of the prime factors in \( 270 \) are:

- \( 2^1 \) (odd)
- \( 3^3 \) (odd)
- \( 5^1 \) (odd)

To make \( 270N \) a perfect square, \( N \) must provide the necessary factors to make all exponents even.

### Step 4: Determine the necessary factors from \( N \)
To make the exponents even:
- We need at least one \( 2 \) from \( N \) to make \( 2^1 \) into \( 2^2 \).
- We need at least one \( 3 \) from \( N \) to make \( 3^3 \) into \( 3^4 \).
- We need at least one \( 5 \) from \( N \) to make \( 5^1 \) into \( 5^2 \).

Thus, \( N \) must be at least \( 2 \times 3 \times 5 = 30 \) to ensure all prime factors have even exponents.

### Step 5: Check values of \( N \) from 30 to 37
Now, we can check the values of \( N \) from 30 to 37 to see which makes \( 270N \) a perfect square.

- **For \( N = 30 \)**:
\[ 270 \times 30 = 8100 \quad (\text{which is } 90^2) \]

- **For \( N = 31 \)**:
\[ 270 \times 31 = 8370 \quad (\text{not a perfect square}) \]

- **For \( N = 32 \)**:
\[ 270 \times 32 = 8640 \quad (\text{not a perfect square}) \]

- **For \( N = 33 \)**:
\[ 270 \times 33 = 8910 \quad (\text{not a perfect square}) \]

- **For \( N = 34 \)**:
\[ 270 \times 34 = 9180 \quad (\text{not a perfect square}) \]

- **For \( N = 35 \)**:
\[ 270 \times 35 = 9450 \quad (\text{not a perfect square}) \]

- **For \( N = 36 \)**:
\[ 270 \times 36 = 9720 \quad (\text{not a perfect square}) \]

- **For \( N = 37 \)**:
\[ 270 \times 37 = 9990 \quad (\text{not a perfect square}) \]

### Conclusion
The only value of \( N \) that makes \( 270N \) a perfect square and a four-digit number is:

\[ \boxed{30} \]