How many factors of the number 4000 are perfect squares ?

Options

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

Answer

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

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

- Start by dividing 4000 by 2:
- 4000 ÷ 2 = 2000
- 2000 ÷ 2 = 1000
- 1000 ÷ 2 = 500
- 500 ÷ 2 = 250
- 250 ÷ 2 = 125
- Now, 125 is not divisible by 2, so we divide by 5:
- 125 ÷ 5 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1

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

### Step 2: Understanding Perfect Squares
A perfect square is a number that can be expressed as the square of an integer. For a factor to be a perfect square, all the exponents in its prime factorization must be even.

### Step 3: Finding the Exponents
From the prime factorization \( 2^5 \times 5^3 \):
- The exponent of 2 is 5, which can contribute even exponents: 0, 2, 4 (3 options).
- The exponent of 5 is 3, which can contribute even exponents: 0, 2 (2 options).

### Step 4: Calculating the Total Number of Perfect Square Factors
To find the total number of perfect square factors, we multiply the number of options for each prime factor:
- For \( 2^5 \): 3 options (0, 2, 4)
- For \( 5^3 \): 2 options (0, 2)

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

### Conclusion
The number of factors of 4000 that are perfect squares is **6**.

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