Math 2.0

Number theory Proposition

Congruence is compatible with multiplication

Builds on Congruence modulo n

Statement

Let nn be a positive integer, and let aa, bb, cc and dd be integers. If a≡b(modn)a \equiv b \pmod n and c≡d(modn)c \equiv d \pmod n, then ac≡bd(modn)a c \equiv b d \pmod n.

Remarks

The proof: ac−bd=a(c−d)+d(a−b)a c - b d = a (c - d) + d (a - b), and nn divides both terms. With compatibility with addition, this result shows that the multiplication of integers modulo n is well defined. By induction, it gives compatibility with powers.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionCongruence is compatible with multiplication, statement, no evidence levelCongruence is compati…statement
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  1. Divisibility definition
  2. Congruence modulo n definition
  3. Congruence is compatible with multiplication statement

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Modular_arithmetic, Basic properties
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