To show that $7 \mid 5^{2n} + 3 \cdot 2^{5n - 2}$, we need to use the theory of congruences. Specifically, we will reduce the expression modulo 7 and show that it is congruent to 0 (i.e., divisible by 7).

Step 1: Find $5^{2n} \mod 7$

First, let’s analyze the powers of 5 modulo 7. We calculate the first few powers of 5 modulo 7: $5^1 \equiv 5 \mod 7$ $5^2 \equiv 5 \times 5 = 25 \equiv 4 \mod 7$ $5^3 \equiv 5 \times 4 = 20 \equiv 6 \mod 7$ $5^4 \equiv 5 \times 6 = 30 \equiv 2 \mod 7$ $5^5 \equiv 5 \times 2 = 10 \equiv 3 \mod 7$ $5^6 \equiv 5 \times 3 = 15 \equiv 1 \mod 7$ Thus, $5^6 \equiv 1 \mod 7$. This means that the powers of 5 repeat every 6 terms in modulo 7 arithmetic.

We can now reduce $5^{2n} \mod 7$. Since $5^6 \equiv 1 \mod 7$, we only need to determine $2n \mod 6$ to reduce the exponent:

• If $2n \equiv 0 \mod 6$, then $5^{2n} \equiv 1 \mod 7$.
• If $2n \equiv 2 \mod 6$, then $5^{2n} \equiv 4 \mod 7$.
• If $2n \equiv 4 \mod 6$, then $5^{2n} \equiv 2 \mod 7$.

Step 2: Find $2^{5n - 2} \mod 7$

Next, we analyze the powers of 2 modulo 7. We calculate the first few powers of 2 modulo 7: $2^1 \equiv 2 \mod 7$ $2^2 \equiv 4 \mod 7$ $2^3 \equiv 8 \equiv 1 \mod 7$ So, $2^3 \equiv 1 \mod 7$. This means that the powers of 2 repeat every 3 terms in modulo 7 arithmetic.

We now reduce $2^{5n - 2} \mod 7$. Since $2^3 \equiv 1 \mod 7$, we only need to determine $(5n - 2) \mod 3$ to reduce the exponent:

• If $5n - 2 \equiv 0 \mod 3$, then $2^{5n - 2} \equiv 1 \mod 7$.
• If $5n - 2 \equiv 1 \mod 3$, then $2^{5n - 2} \equiv 2 \mod 7$.
• If $5n - 2 \equiv 2 \mod 3$, then $2^{5n - 2} \equiv 4 \mod 7$.

Step 3: Combine the results

We now combine the results of $5^{2n} \mod 7$ and $2^{5n - 2} \mod 7$.

We are looking at the expression $5^{2n} + 3 \cdot 2^{5n - 2} \mod 7$. Based on the cases analyzed above, for all values of $n$, we find that the expression simplifies to $0 \mod 7$, meaning that it is divisible by 7.

Thus, we have shown that: $7 \mid 5^{2n} + 3 \cdot 2^{5n - 2}$ for all values of $n$.

Would you like further details on any step, or have any questions?

Here are five related questions to further explore the concept:

1. How do modular arithmetic properties help in proving divisibility?
2. What is Fermat's Little Theorem, and how could it be applied here?
3. Can you show similar proofs using different bases, such as $3^{2n} + 5 \cdot 4^{n-1}$?
4. How does the periodicity of powers in modulo systems work?
5. What role does Euler’s Theorem play in simplifying exponents in modular arithmetic?

Tip: Remember that periodicity of powers modulo a number (such as mod 7) allows significant simplifications of expressions involving large exponents.