In aptitude (remainder and modular arithmetic), when the expression 7^21 + 7^22 + 7^23 + 7^24 is divided by 25, what remainder is obtained? Use properties of powers and modular cycles for efficient calculation.

Introduction / Context:
This question is a typical modular arithmetic problem where we must find the remainder when a large power expression is divided by a given number. Direct calculation of 7^21 and higher powers is impractical, so we use patterns in powers of 7 modulo 25. Recognising repeating cycles in the remainders is the main skill tested here. Such problems are widely used in quantitative aptitude and number theory sections of competitive exams.

Given Data / Assumptions:
- Expression: 7^21 + 7^22 + 7^23 + 7^24.
- Divisor: 25.
- We need the remainder when the entire sum is divided by 25.
- Standard rules of modular arithmetic apply: (a + b) mod m = [(a mod m) + (b mod m)] mod m, and similar for products.

Concept / Approach:
The approach is to find the pattern in powers of 7 modulo 25. We compute 7^1 mod 25, 7^2 mod 25, and so on until we see a cycle. Once a cycle is identified, we reduce the exponents 21, 22, 23, and 24 modulo the cycle length to find equivalent smaller exponents. Then we compute the corresponding remainders, add them, and finally find the remainder when that sum is divided by 25.

Step-by-Step Solution:
Step 1: Compute initial powers of 7 modulo 25.7^1 mod 25 = 7.7^2 = 49; 49 mod 25 = 49 - 25 * 1 = 24.7^3 = 7^2 * 7 = 24 * 7 = 168; 168 mod 25 = 168 - 25 * 6 = 168 - 150 = 18.7^4 = 7^3 * 7 = 18 * 7 = 126; 126 mod 25 = 126 - 25 * 5 = 126 - 125 = 1.Step 2: We observe that 7^4 mod 25 = 1. This means powers of 7 repeat every 4 exponents modulo 25.Step 3: Therefore, 7^n mod 25 depends on n mod 4.Step 4: Compute remainders of exponents modulo 4: 21 mod 4 = 1, 22 mod 4 = 2, 23 mod 4 = 3, 24 mod 4 = 0.Step 5: So 7^21 mod 25 = 7^1 mod 25 = 7.Step 6: 7^22 mod 25 = 7^2 mod 25 = 24.Step 7: 7^23 mod 25 = 7^3 mod 25 = 18.Step 8: 7^24 mod 25 = 7^4 mod 25 = 1.Step 9: Now sum these remainders: 7 + 24 + 18 + 1 = 50.Step 10: Finally, 50 mod 25 = 0.

Verification / Alternative check:
An alternative view is to factor out 7^21: 7^21(1 + 7 + 7^2 + 7^3). Modulo 25, we already know that 7^4 ≡ 1, so 7^0 ≡ 1, 7^1 ≡ 7, 7^2 ≡ 24, 7^3 ≡ 18, and 1 + 7 + 24 + 18 = 50 ≡ 0 mod 25. Therefore, 7^21 * 0 ≡ 0 mod 25, confirming that the entire expression leaves remainder zero when divided by 25.

Why Other Options Are Wrong:
- 5, 7, 10, 17: These values might appear if one miscalculates powers or stops the remainder calculation prematurely. However, once the full pattern and sum are handled correctly, the remainder is exactly 0. None of these other values can be justified by correct modular arithmetic steps.

Common Pitfalls:
Common errors include computing large powers directly without using cycles, leading to arithmetic mistakes, or misidentifying the cycle length. Some students incorrectly assume a cycle length of 5 or 10 without checking. Others may forget to reduce each power mod 25 before summing, causing overflow and confusion. Always establish the cycle carefully and then apply it to each exponent.

Final Answer:
The remainder obtained when 7^21 + 7^22 + 7^23 + 7^24 is divided by 25 is 0.