SUMMARY
This discussion focuses on solving congruences using Fermat's Little Theorem. The first problem involves finding 'a' such that \(0 < a < 73\) and \(a \equiv 9^{794} \mod 73\). The solution utilizes the theorem to simplify the exponent, concluding that \(9^{794} \equiv 9^2 \mod 73\). The second problem, \(x^{86} \equiv 6 \mod 29\), is approached by recognizing that \(x^{28} \equiv 1 \mod 29\), leading to the simplification \(x^{86} \equiv x^2 \equiv 6 \mod 29\).
PREREQUISITES
- Understanding of modular arithmetic
- Fermat's Little Theorem
- Basic exponentiation techniques in modular contexts
- Knowledge of prime numbers and their properties
NEXT STEPS
- Study applications of Fermat's Little Theorem in cryptography
- Learn advanced techniques in modular exponentiation
- Explore the Chinese Remainder Theorem for solving systems of congruences
- Investigate the properties of prime numbers in number theory
USEFUL FOR
Mathematicians, students studying number theory, and anyone interested in solving congruences and applying Fermat's Little Theorem in practical scenarios.
