We are tasked with finding the remainder when $7^{21} + 7^{22} + 7^{23} + 7^{24}$ is divided by 25.

To solve this efficiently, we can use modular arithmetic. Specifically, we will find $7^n \mod 25$ for $n = 21, 22, 23, 24$ and then sum the results, taking the remainder modulo 25.

Step 1: Finding the Powers of 7 Modulo 25

We'll first compute powers of 7 modulo 25. Since powers of a number modulo another often exhibit a cyclic pattern, we'll begin by calculating:

$7^1 \mod 25 = 7$ $7^2 \mod 25 = 7 \times 7 = 49 \mod 25 = 24$ $7^3 \mod 25 = 7 \times 24 = 168 \mod 25 = 18$ $7^4 \mod 25 = 7 \times 18 = 126 \mod 25 = 1$

Step 2: Identify the Cycle

At this point, we can see that after every four exponents, the powers of 7 modulo 25 repeat. This is because $7^4 \equiv 1 \mod 25$. Therefore, we have a cycle:

• $7^1 \mod 25 = 7$
• $7^2 \mod 25 = 24$
• $7^3 \mod 25 = 18$
• $7^4 \mod 25 = 1$

Step 3: Find $7^{21}, 7^{22}, 7^{23}, 7^{24} \mod 25$

Since the powers of 7 repeat every 4 terms, we can reduce the exponents $21, 22, 23, 24$ modulo 4: $21 \mod 4 = 1, \quad 22 \mod 4 = 2, \quad 23 \mod 4 = 3, \quad 24 \mod 4 = 0$

Thus:

• $7^{21} \equiv 7^1 \mod 25 = 7$
• $7^{22} \equiv 7^2 \mod 25 = 24$
• $7^{23} \equiv 7^3 \mod 25 = 18$
• $7^{24} \equiv 7^4 \mod 25 = 1$

Step 4: Sum and Find the Remainder

Now we sum the results: $7 + 24 + 18 + 1 = 50$

Finally, find the remainder when 50 is divided by 25: $50 \mod 25 = 0$

Conclusion:

The remainder when $7^{21} + 7^{22} + 7^{23} + 7^{24}$ is divided by 25 is $\boxed{0}$.

Would you like further details on any part of this solution?

Related Questions:

1. What is the general technique for finding powers of numbers modulo a given number?
2. How can Fermat's Little Theorem be applied in modular arithmetic problems?
3. What is the role of cyclic patterns in modular arithmetic?
4. How would this solution change if we had a different modulus, say 26 instead of 25?
5. What is Euler’s theorem, and how does it relate to modular exponentiation?

Tip:

When dealing with large exponents in modular arithmetic, always look for patterns or cycles to simplify your calculations.