How many factors of the number 4000 are perfect squares ?

Options

A. 7
B. 5
C. 6
D. 4

Answer

To find how many factors of the number 4000 are perfect squares, we can follow these steps:

### Step 1: Prime Factorization of 4000
First, we need to find the prime factorization of 4000.

4000 can be broken down as follows:
- 4000 = 4 × 1000
- 4 = 2^2
- 1000 = 10 × 100 = 10 × 10 × 10 = (2 × 5) × (10 × 10) = (2 × 5) × (2 × 5) × (2 × 5) = 2^3 × 5^3

Now, combining these:
- 4000 = 2^2 × (2^3 × 5^3) = 2^(2+3) × 5^3 = 2^5 × 5^3

So, the prime factorization of 4000 is:
\[ 4000 = 2^5 \times 5^3 \]

### Step 2: Finding the Total Number of Factors
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 total number of factors is given by:
\[ (e_1 + 1)(e_2 + 1)...(e_k + 1) \]

For 4000, we have:
- \( e_1 = 5 \) (for \( 2^5 \))
- \( e_2 = 3 \) (for \( 5^3 \))

So, the total number of factors is:
\[ (5 + 1)(3 + 1) = 6 \times 4 = 24 \]

### Step 3: Finding Perfect Square Factors
A factor is a perfect square if all the powers in its prime factorization are even.

For \( 4000 = 2^5 \times 5^3 \):
- The possible even powers for \( 2 \) are \( 0, 2, 4 \) (3 options)
- The possible even powers for \( 5 \) are \( 0, 2 \) (2 options)

Now, we multiply the number of choices:
\[ \text{Number of perfect square factors} = \text{(choices for 2)} \times \text{(choices for 5)} = 3 \times 2 = 6 \]

### Conclusion
Thus, the number of factors of 4000 that are perfect squares is **6**.