The problem asks for the sum of all integers between 1 and 25 that are not factors of $N$, where:

$N = 242024 + 242025$

Step 1: Simplify $N$

$N = 242024 + 242025 = 484049$

Step 2: Determine the factors of $N$ between 1 and 25

We need to check which integers from 1 to 25 divide $N = 484049$ without a remainder.

Step 3: Verify $N$'s divisors between 1 and 25

The prime factorization or divisors can be tested systematically. However, for practical purposes:

1. $N = 484049$ is $695^2$, a perfect square. Hence, its divisors are based on $695^2$.

We note factors! ProceedLet’s carefully analyze the given problem:

Step 1: Compute $N$

$N = 242024 + 242025 = 484049.$

This simplifies $N$ as:

$N = 695^2 \, (\text{because } 695 \times 695 = 484049).$

Step 2: Factors of $N$

The number $N = 695^2$ has divisors that include 1, 695, and 484049, among others. To find factors of $N$ specifically between 1 and 25, we test each integer from 1 to 25 to see if it divides $N$ without remainder.

Step 3: Divisors of $N$ between 1 and 25

We systematically test each number from 1 to 25:

• $1$: Always a factor.
• $2, 3, 4, 5, 6, \dots, 24, 25$: None of these are factors because $695$ is odd, and its prime factors are not divisible by 2, 3, or 5. Therefore, only 1 is a factor of $N$ in this range.

Step 4: Integers NOT factors of $N$

The integers between 1 and 25 are: $\{1, 2, 3, 4, \dots, 25\}.$

Since only 1 is a factor, the numbers not factors of $N$ are: $\{2, 3, 4, 5, \dots, 25\}.$

Step 5: Sum of integers not factors of $N$

The sum of integers from 1 to 25 is: $\text{Sum} = \frac{25 \times 26}{2} = 325.$

Subtract the factor $1$: $325 - 1 = 324.$

Final Answer:

The sum of all integers between 1 and 25 that are not factors of $N$ is: $\boxed{324}.$

Would you like further clarification or a deeper breakdown?

Here are 5 related questions for exploration:

1. How can you systematically determine all factors of a number?
2. Why does the prime factorization of a number help identify divisors?
3. What is the significance of $695^2$ in the factorization process?
4. How would the sum change if the range of integers was extended to 50?
5. How do perfect squares impact the calculation of factors?

Tip: Always check for prime factorization when analyzing divisors—it simplifies the process significantly!