SUMMARY
The last two digits of the expression $2^{2n}(2^{2n+1}-1)$, where n is an odd positive integer, can be determined using modular arithmetic. Specifically, the expression simplifies to $0 \mod 4$ and $1 \mod 25$, leading to the conclusion that the last two digits are 00. This result is consistent across various odd values of n, confirming the robustness of the conclusion.
PREREQUISITES
- Understanding of modular arithmetic
- Familiarity with binary and decimal number systems
- Knowledge of exponentiation properties
- Basic number theory concepts
NEXT STEPS
- Research modular arithmetic techniques for large numbers
- Explore properties of powers of 2 in different bases
- Study the Chinese Remainder Theorem for solving congruences
- Learn about binary representation and its applications in number theory
USEFUL FOR
Mathematicians, students studying number theory, and anyone interested in modular arithmetic and its applications in solving problems involving powers and residues.