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The problem you have shared consists of the following tasks:

**Given:**
\[
A = 5^{50} - 5^{48} + 5^{46} - 5^{44} + \cdots + 5^6 - 5^4 + 5^2 - 1
\]

### (a) Calculate \( A \)

### (b) Find the natural number \( n \) such that \( 26A + 1 = 5^n \)

### (c) Find the remainder when \( A \) is divided by 100

---

Let me start by solving part (a) to calculate \( A \).

### Part (a) - Simplifying \( A \):

The expression for \( A \) follows a pattern of alternating signs, which can be factored.

We observe that:
\[
A = (5^{50} - 5^{48}) + (5^{46} - 5^{44}) + \cdots + (5^2 - 1)
\]
This can be written as:
\[
A = (5^{48}(5^2 - 1)) + (5^{44}(5^2 - 1)) + \cdots + (5^2 - 1)
\]

Since \( 5^2 - 1 = 24 \), we can factor out 24:
\[
A = 24(5^{48} + 5^{44} + \cdots + 1)
\]
Now, I will calculate the sum of the powers of 5 and then multiply it by 24.

Let's compute this next.The value of \( A \) is:

\[
A = 85,401,771,125,012,041,571,048,589,853,140,024
\]

### Moving on to part (b):

We need to find a natural number \( n \) such that \( 26A + 1 = 5^n \). Let's calculate \( 26A + 1 \) and determine whether it is a power of 5.The value of \( 26A + 1 \) is:

\[
26A + 1 = 2,220,446,049,250,313,080,847,263,336,181,640,625
\]

This number is equal to \( 5^{44} \), meaning \( n = 44 \).

### Part (b) answer: \( n = 44 \).

---

### Part (c) - Find the remainder when \( A \) is divided by 100:

Let's compute \( A \mod 100 \).It seems like I can't do more advanced data analysis right now. Please try again later.

Would you like to proceed with any other questions or calculations?

Here are five related questions you might be interested in:
1. How can we simplify large exponents further?
2. How do alternating sums affect expressions like the one in this problem?
3. What is the general method to find remainders using modular arithmetic?
4. What are the steps to express numbers in terms of powers?
5. How do factorizations help in simplifying large sums?

**Tip:** When dealing with very large numbers, modular arithmetic can help reduce complexity, especially in remainder calculations.

Given A = 5^50 - 5^48 + 5^46 - 5^44 + ... + 5^6 - 5^4 + 5^2 - 1.
(a) Calculate A.
(b) Find the natural number n such that 26A + 1 = 5^n.
(c) Find the remainder when A is divided by 100.

This problem focuses on calculating a sum of exponents involving powers of 5, finding a natural number n where 26A + 1 = 5^n, and determining the remainder when A is divided by 100. Suitable for advanced algebra and number theory students.

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