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Number-Theory · Modular-Arithmetic

Inverse of 4 mod 9

4,\ 9

Inyathelo ngenyathelo

  1. 4x \equiv 1 \pmod{9}

    We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.

  2. 9 = 2 \times 4 + 1

    Euclid step.

  3. 4 = 4 \times 1 + 0

    Euclid step.

  4. 4 \times 7 = 28 \equiv 1 \pmod{9}

    Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.

Bonisa impendulo
4^{-1} \equiv 7 \pmod{9}