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The image provided contains a math problem, which asks for the remainder (modulus operation) when a certain expression involving powers of 2024 is divided by 17.

The expression provided is:

\[
N = 2024^{1} + 2024^{2} + 2024^{3} + 2024^{4} + \dots + 2024^{99}
\]

And the task is to find the remainder when \( N \) is divided by 17, or:

\[
N \mod 17
\]

This is a modulus problem involving powers of a number reduced by a modulus. To solve this, we can use properties of modular arithmetic and periodicity (since \( 2024 \mod 17 \) is congruent to a smaller number mod 17). Here's a step-by-step outline of the solution approach:

1. Find \( 2024 \mod 17 \).
2. Use this result to find the powers \( 2024^1 \mod 17, 2024^2 \mod 17, \dots, 2024^{99} \mod 17 \).
3. Sum all the results and take the modulus again.

Let me calculate that for you.We have:

\[
2024 \mod 17 = 1
\]

This means that \( 2024 \equiv 1 \mod 17 \). 

Now, any power of 2024 will also be congruent to 1 mod 17. Specifically:

\[
2024^k \mod 17 = 1^k \mod 17 = 1
\]

Thus, the expression \( N = 2024^1 + 2024^2 + \dots + 2024^{99} \) becomes:

\[
N \equiv 1 + 1 + 1 + \dots + 1 \mod 17
\]

Since there are 99 terms, this simplifies to:

\[
N \equiv 99 \mod 17
\]

Let me compute \( 99 \mod 17 \).We have:

\[
99 \mod 17 = 14
\]

Therefore, the remainder when \( N = 2024^1 + 2024^2 + \dots + 2024^{99} \) is divided by 17 is:

\[
N \mod 17 = 14
\]

Would you like a more detailed explanation of the steps? Here are 5 related questions to explore:

1. How does modular arithmetic work with powers of numbers?
2. What are other ways to simplify modular exponentiation problems?
3. Why does the periodicity of numbers modulo a divisor occur?
4. How can Fermat's Little Theorem help in similar modulus problems?
5. What is the general approach for solving large power modulus problems?

**Tip:** When dealing with powers and modular arithmetic, look for patterns and periodicity to simplify calculations!

Given the expression N = 2024^1 + 2024^2 + 2024^3 + ... + 2024^99, find the remainder when N is divided by 17.

Learn how to find the remainder of the expression N = 2024^1 + 2024^2 + ... + 2024^99 when divided by 17 using modular arithmetic and periodic properties. Suitable for high school students.

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