SUMMARY
The discussion centers on solving the modular equation 7x ≡ 3 (mod 15). It is established that the equation does not hold true for all integer values of x, as demonstrated by the counterexample where x = 3 yields 7*3 = 21 ≡ 6 (mod 15). The correct approach involves finding specific values of x that satisfy the equation, which can be reformulated as x = (3/7) (mod 15). The solution requires determining the modular inverse of 7 modulo 15, which is found to be 13, leading to the conclusion that 3/7 (mod 15) can be computed accordingly.
PREREQUISITES
- Understanding of modular arithmetic
- Familiarity with modular inverses
- Basic algebraic manipulation skills
- Knowledge of congruences and their properties
NEXT STEPS
- Study how to compute modular inverses using the Extended Euclidean Algorithm
- Learn about solving linear congruences
- Explore applications of modular arithmetic in cryptography
- Practice additional problems involving modular equations
USEFUL FOR
Students studying number theory, mathematicians interested in modular arithmetic, and anyone solving linear congruences in algebra.