SUMMARY
The calculation of the last digit of 10^1000 modulo 8 is confirmed to be 0. The discussion outlines the steps taken, starting with the equivalence 10^2 ≡ 4 (mod 8) and progressing to (10^2)^500 ≡ 4^500 (mod 8). It further establishes that 4^2 ≡ 0 (mod 8), leading to the conclusion that (4^2)^250 ≡ 0 (mod 8). Therefore, the last digit of 10^1000 mod 8 is definitively 0.
PREREQUISITES
- Understanding of modular arithmetic
- Familiarity with exponentiation rules in modular contexts
- Knowledge of equivalence relations in mathematics
- Basic algebraic manipulation skills
NEXT STEPS
- Study modular exponentiation techniques
- Explore properties of congruences in number theory
- Learn about applications of modular arithmetic in cryptography
- Investigate the Chinese Remainder Theorem for solving modular equations
USEFUL FOR
Students in mathematics, particularly those studying number theory, educators teaching modular arithmetic, and anyone interested in mathematical proofs and problem-solving techniques.