How many of the factors of 512 are perfect squares?

Options

A. 3
B. 5
C. 6
D. 4

Answer

To find how many of the factors of 512 are perfect squares, we can follow these steps:

### Step 1: Find the prime factorization of 512
First, we need to express 512 as a product of its prime factors.
- 512 can be divided by 2 repeatedly:
- 512 ÷ 2 = 256
- 256 ÷ 2 = 128
- 128 ÷ 2 = 64
- 64 ÷ 2 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1

Thus, the prime factorization of 512 is:
\[ 512 = 2^9 \]

### Step 2: Determine the total number of factors of 512
To find the total number of factors of a number from its prime factorization, we use the formula:
If \( n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k} \), then the number of factors \( T(n) \) is given by:
\[ T(n) = (e_1 + 1)(e_2 + 1)...(e_k + 1) \]

For 512:
- Here, \( e_1 = 9 \) (since \( 512 = 2^9 \)).
- Therefore, the total number of factors is:
\[ T(512) = (9 + 1) = 10 \]

### Step 3: Identify the perfect square factors
A factor is a perfect square if all the exponents in its prime factorization are even.

The exponent of 2 in the factorization of 512 can be any even number from 0 to 9:
- Possible even exponents: 0, 2, 4, 6, 8

### Step 4: Count the perfect square factors
The possible even exponents are:
- 0 (which corresponds to \( 2^0 = 1 \))
- 2 (which corresponds to \( 2^2 = 4 \))
- 4 (which corresponds to \( 2^4 = 16 \))
- 6 (which corresponds to \( 2^6 = 64 \))
- 8 (which corresponds to \( 2^8 = 256 \))

Thus, the perfect square factors of 512 are:
1, 4, 16, 64, and 256.

### Step 5: Conclusion
The total number of perfect square factors of 512 is 5.

### Final Answer
**5**

---