How many distinct factors of 1600 are perfect squares?

Options

A. 12
B. 10
C. 9
D. 8

Answer

To find how many distinct factors of 1600 are perfect squares, we can follow these steps:

### Step 1: Prime Factorization of 1600
First, we need to find the prime factorization of 1600.

- Start by dividing 1600 by 2 (the smallest prime number):
- 1600 ÷ 2 = 800
- 800 ÷ 2 = 400
- 400 ÷ 2 = 200
- 200 ÷ 2 = 100
- 100 ÷ 2 = 50
- 50 ÷ 2 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1

So, the prime factorization of 1600 is:
\[ 1600 = 2^5 \times 5^2 \]

### Step 2: Identifying Perfect Square Factors
A perfect square factor must have even powers in its prime factorization.

- The general form of a factor of 1600 can be expressed as \( 2^a \times 5^b \), where:
- \( 0 \leq a \leq 5 \)
- \( 0 \leq b \leq 2 \)

For the factor to be a perfect square:
- \( a \) must be even: possible values are 0, 2, 4 (3 options)
- \( b \) must be even: possible values are 0, 2 (2 options)

### Step 3: Counting the Perfect Square Factors
To find the total number of perfect square factors, we multiply the number of choices for \( a \) and \( b \):

- Number of choices for \( a \) (even powers of 2): 3 (0, 2, 4)
- Number of choices for \( b \) (even powers of 5): 2 (0, 2)

Thus, the total number of distinct perfect square factors is:
\[ 3 \times 2 = 6 \]

### Conclusion
The number of distinct factors of 1600 that are perfect squares is **6**.

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