1008 divided by which single digit number gives a perfect square ?

Options

A. 9
B. 4
C. 8
D. 7

Answer

To find which single-digit number divides 1008 to give a perfect square, we can follow these steps:

### Step 1: Factor the number 1008
We start by finding the prime factorization of 1008.

1. Divide by 2:
- 1008 ÷ 2 = 504
2. Divide by 2 again:
- 504 ÷ 2 = 252
3. Divide by 2 again:
- 252 ÷ 2 = 126
4. Divide by 2 again:
- 126 ÷ 2 = 63
5. Divide by 3:
- 63 ÷ 3 = 21
6. Divide by 3 again:
- 21 ÷ 3 = 7
7. Finally, we have 7, which is a prime number.

So, the prime factorization of 1008 is:
\[ 1008 = 2^4 \times 3^2 \times 7^1 \]

### Step 2: Analyze the factors for perfect squares
A perfect square has all prime factors raised to even powers. In the factorization \( 2^4 \times 3^2 \times 7^1 \):
- The power of 2 is 4 (even).
- The power of 3 is 2 (even).
- The power of 7 is 1 (odd).

### Step 3: Determine which single-digit number to divide by
To make the power of 7 even, we need to divide by 7. This will remove the odd power of 7, resulting in:
\[ \frac{1008}{7} = 144 \]

### Step 4: Check if the result is a perfect square
Now, we check if 144 is a perfect square:
- The prime factorization of 144 is:
\[ 144 = 12 \times 12 = 2^4 \times 3^2 \]
- Both powers are even, confirming that 144 is indeed a perfect square.

### Conclusion
Thus, the single-digit number that divides 1008 to give a perfect square is **7**.

---