How does division (fraction computation) work in modular arithmetic?

Division mod N is multiplication by the inverse: $\frac{x}{y} \mod N \equiv (x \cdot y^{-1}) \mod N$. You NEVER actually divide — you find the inverse and multiply.

Formula: $z \equiv \frac{x}{y} \mod N \equiv (x \cdot y^{-1}) \mod N$

Verification: $x \equiv (z \cdot y) \mod N$

Example: $\frac{5}{7} \mod 9$

Method 1 — Find inverse first:

• Find $7^{-1} \mod 9$: try $t$ where $t \cdot 7 \equiv 1 \mod 9$ → $t = 4$ (since $4 \cdot 7 = 28 \equiv 1 \mod 9$)
• Compute $5 \cdot 4 = 20 \equiv 2 \mod 9$

Method 2 — Direct search:

• Find $z$ where $z \cdot 7 \equiv 5 \mod 9$ → $z = 2$ (since $2 \cdot 7 = 14 \equiv 5 \mod 9$)

Important warnings:

• Never cancel/simplify fractions in modular arithmetic — cancellation rules from regular math don't apply!
• Standard calculators (TI-89 etc.) compute $\frac{5}{7} \mod 9$ incorrectly — they don't understand modular fractions
• Use tools like Wolfram Alpha or Maple for reliable results

Go deeper:

• Modular multiplicative inverse (Wikipedia) — dividing mod N = multiplying by the inverse.