For what value of N, the number 34N will be a perfect cube, where 34N is a 3-digit number ?

Options

A. 2
B. 3
C. 4
D. 5

Answer

To determine the value of N such that the number 34N is a perfect cube and is a three-digit number, we can follow these steps:

### Step 1: Understand the range of three-digit numbers
Three-digit numbers range from 100 to 999. Therefore, we need to find values of N such that 34N falls within this range.

### Step 2: Set up the inequality
We can express this as:
\[ 100 \leq 34N \leq 999 \]

### Step 3: Solve for N
To find the range of N, we can divide the entire inequality by 34:
\[ \frac{100}{34} \leq N \leq \frac{999}{34} \]

Calculating the bounds:
- Lower bound:
\[ \frac{100}{34} \approx 2.94 \]
- Upper bound:
\[ \frac{999}{34} \approx 29.32 \]

Since N must be a single digit (0-9), we can round the lower bound up to 3 (since N must be an integer).

### Step 4: List possible values of N
The possible integer values of N within the range are 3, 4, 5, 6, 7, 8, and 9.

### Step 5: Check for perfect cubes
Next, we need to check which of the numbers 340, 341, 342, ..., 349 (which corresponds to N = 0, 1, 2, ..., 9) are perfect cubes.

Calculating the cubes of integers:
- \( 1^3 = 1 \)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)
- \( 5^3 = 125 \)
- \( 6^3 = 216 \)
- \( 7^3 = 343 \)
- \( 8^3 = 512 \)
- \( 9^3 = 729 \)
- \( 10^3 = 1000 \) (not a three-digit number)

### Step 6: Identify the perfect cube
From the above calculations, we see that:
- \( 343 \) is a perfect cube (which is \( 7^3 \)).

### Step 7: Determine the value of N
Now, we check which value of N gives us 343:
- \( 34N = 343 \)
- Solving for N:
\[ N = \frac{343}{34} \approx 10.09 \] (not valid since N must be a single digit)

Next, we check \( N = 3 \):
- \( 34 \times 3 = 102 \) (not a perfect cube)
- Check \( N = 4 \):
- \( 34 \times 4 = 136 \) (not a perfect cube)
- Check \( N = 5 \):
- \( 34 \times 5 = 170 \) (not a perfect cube)
- Check \( N = 6 \):
- \( 34 \times 6 = 204 \) (not a perfect cube)
- Check \( N = 7 \):
- \( 34 \times 7 = 238 \) (not a perfect cube)
- Check \( N = 8 \):
- \( 34 \times 8 = 272 \) (not a perfect cube)
- Check \( N = 9 \):
- \( 34 \times 9 = 306 \) (not a perfect cube)

### Conclusion
The only value of N that makes 34N a perfect cube is when N = 3, resulting in 343, which is \( 7^3 \).

### Final Answer
Thus, the value of N is **3**.