a number N is between 2 to the fifth power and 3 to the fourth power
. How many possible values of N are there that has even number of factors?

We are given that a number \( N \) lies between \( 2^5 \) and \( 3^4 \), and we need to determine how many possible values of \( N \) have an even number of factors.

### Step 1: Calculate \( 2^5 \) and \( 3^4 \)
\[
2^5 = 32
\]
\[
3^4 = 81
\]
So, the number \( N \) lies between 32 and 81.

### Step 2: Determine the numbers between 32 and 81
The integers between 32 and 81 are:
\[
33, 34, 35, \ldots, 80
\]
These numbers are from 33 to 80, inclusive, which gives us:
\[
80 - 33 + 1 = 48 \text{ possible values of } N.
\]

### Step 3: Determine when a number has an even number of factors
A number has an **even** number of factors unless it is a **perfect square**. This is because the factors of a number usually come in pairs (e.g., for 12, the factors are \( 1 \times 12 \), \( 2 \times 6 \), \( 3 \times 4 \)), but for a perfect square, one factor is repeated (e.g., for 36, the factors are \( 1 \times 36 \), \( 2 \times 18 \), \( 3 \times 12 \), \( 4 \times 9 \), \( 6 \times 6 \)).

### Step 4: Identify the perfect squares between 32 and 81
We find the perfect squares between 32 and 81:
\[
6^2 = 36, \quad 7^2 = 49, \quad 8^2 = 64, \quad 9^2 = 81
\]
So the perfect squares in this range are 36, 49, 64, and 81.

### Step 5: Count the numbers with an even number of factors
The total number of integers between 33 and 80 is 48. Out of these, 4 are perfect squares (36, 49, 64, 81), which have an odd number of factors. The remaining numbers will have an even number of factors:
\[
48 - 4 = 44
\]

### Final Answer:
There are **44 numbers** between 32 and 81 that have an even number of factors.

---

Would you like further details or have any questions? Here are some related questions:

1. How do you calculate the number of factors of a given number?
2. What is the significance of perfect squares in factorization?
3. Can a number have both an odd and an even number of factors?
4. How would the answer change if the range for \( N \) was extended?
5. What is the connection between prime factorization and the number of divisors?

**Tip:** The number of divisors of a number can be found using its prime factorization!

A number N is between 2 to the fifth power and 3 to the fourth power. How many possible values of N are there that have an even number of factors?

Discover how to find how many numbers between 32 and 81 have an even number of factors. This problem covers concepts from number theory, including factors and perfect squares, suitable for students in Grades 8-10.

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