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.

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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!