The key to solving this problem lies in understanding the properties of divisors and prime factorization. Let's break down the solution step by step:

Step 1: Finding the prime factorization of N
To find the smallest integer N with exactly 24 divisors, we need to determine its prime factorization. We can represent N as a product of prime factors raised to certain powers. Let's assume the prime factorization of N is:

N = p1^a1 * p2^a2 * p3^a3 * ... * pn^an

Here, p1, p2, p3, ..., pn are prime numbers, and a1, a2, a3, ..., an are positive integers representing the powers.

Step 2: Understanding the number of divisors
The number of divisors a number has can be determined by the powers of its prime factors. If a number has a prime factorization of the form:

N = p1^a1 * p2^a2 * p3^a3 * ... * pn^an

Then the number of divisors of N is given by:

(Number of divisors) = (a1 + 1) * (a2 + 1) * (a3 + 1) * ... * (an + 1)

This formula works because each factor can be chosen independently, and adding 1 accounts for the possibility of not choosing a factor at all.

Step 3: Determining the powers for a given number of divisors
We know that the smallest number with exactly 24 divisors will have the smallest possible powers for its prime factors. To minimize the value of N, we want the powers (a1, a2, a3, ..., an) to be as small as possible.

Step 4: Finding the value of N/40
Now that we have determined the prime factorization of N with the smallest possible powers, we can find the value of N/40.

N/40 = (p1^a1 * p2^a2 * p3^a3 * ... * pn^an) / 40

Since we want to find the smallest possible value of N, we need to minimize the value of N/40. This can be achieved by choosing the smallest possible prime factors and powers.

Step 5: Simplifying N/40
To simplify N/40, we need to determine the prime factorization of 40 and cancel out any common factors between N and 40.

The prime factorization of 40 is: 2^3 * 5^1

N/40 = (p1^a1 * p2^a2 * p3^a3 * ... * pn^an) / (2^3 * 5^1)

By canceling out common factors, we can simplify N/40 further.

Step 6: Finding the final answer
The final answer is the simplified form of N/40, which can be obtained by dividing the powers of the remaining prime factors by the corresponding powers of 2 and 5.

The final answer is '9'