Overview

Intermediate number theory emphasizes modular arithmetic, gcd arguments, and careful factoring.

Key Ideas

• If $a\equiv b\pmod n$, then $a^k\equiv b^k\pmod n$ for integer $k\ge0$.
• The equation $ax+by=c$ has integer solutions iff $\gcd(a,b)\mid c$.
• Use modular constraints to eliminate impossible cases quickly.

Worked Example

Solve $7x\equiv 1\pmod{10}$.

Check small residues: $7\cdot 3=21\equiv 1\pmod{10}$, so $x\equiv 3\pmod{10}$.

Practice Problems