To solve the problem of finding a single-digit number that, when 1008 is divided by it, results in a perfect square, we can follow these steps:
### Step 1: Factorize 1008
First, we need to find the prime factorization of 1008. We can do this by dividing 1008 by the smallest prime numbers until we reach 1.
- Divide by 2:
- 1008 ÷ 2 = 504
- 504 ÷ 2 = 252
- 252 ÷ 2 = 126
- 126 ÷ 2 = 63
- Now, 63 is not divisible by 2, so we move to the next prime number, which is 3:
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- Finally, 7 is a prime number.
Thus, the prime factorization of 1008 is:
\[ 1008 = 2^4 \times 3^2 \times 7^1 \]
### Step 2: Identify the Conditions for a Perfect Square
A number is a perfect square if all the exponents in its prime factorization are even. In our case:
- The exponent of 2 is 4 (even)
- The exponent of 3 is 2 (even)
- The exponent of 7 is 1 (odd)
To make the exponent of 7 even, we need to multiply by 7 (which will increase the exponent of 7 from 1 to 2).
### Step 3: Determine the Single-Digit Divisor
To find a single-digit number that can be used to divide 1008 and result in a perfect square, we can check the single-digit numbers (1 through 9) to see which one, when used to divide 1008, results in a perfect square.
We already established that we need to eliminate the odd exponent of 7. Therefore, we need to divide by 7 to achieve this.
### Step 4: Verify the Result
Now, let's check:
\[ 1008 \div 7 = 144 \]
Next, we check if 144 is a perfect square:
- The prime factorization of 144 is:
\[ 144 = 12 \times 12 = (2^2 \times 3)^2 = 2^4 \times 3^2 \]
- All exponents are even, confirming that 144 is indeed a perfect square.
### Conclusion
Thus, the single-digit number that, when 1008 is divided by it, gives a perfect square is **7**.
