Find the remainder when 29^18 is divided by 26. Choose the correct remainder.

Introduction / Context:
This modular arithmetic question requires reducing a large power modulo 26. Using properties such as Euler’s theorem or pattern cycles helps avoid huge computations and leads quickly to the correct remainder.

Given Data / Assumptions:

• We want r = 29^18 mod 26.
• 29 mod 26 = 3, so 29^18 mod 26 = 3^18 mod 26.
• gcd(3, 26) = 1, enabling Euler’s theorem with phi(26) = 12.

Concept / Approach:
Apply modular reduction of the base and exponent properties. Since 3 and 26 are coprime, 3^12 ≡ 1 (mod 26). Then reduce the exponent 18 using the cycle length 12 to simplify the computation.

Step-by-Step Solution:

Verification / Alternative check:
Direct small-cycle check: powers of 3 mod 26 repeat with period 3 since 3^3 ≡ 1; then 3^18 = (3^3)^6 ≡ 1^6 ≡ 1.

Why Other Options Are Wrong:
29 and 26 are not valid remainders under modulus 26 (26 represents zero). 18 is arbitrary. 0 would imply divisibility, which does not hold.

Common Pitfalls:
Forgetting to reduce the base first or misapplying Euler’s theorem without checking coprimality.

Final Answer:
1