Math 2.0

Number theory Proposition

Congruence is compatible with addition

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 a+c≡b+d(modn)a + c \equiv b + d \pmod n.

Remarks

The proof: (a+c)−(b+d)=(a−b)+(c−d)(a + c) - (b + d) = (a - b) + (c - d), and nn divides both terms. The same holds for subtraction: a−c≡b−d(modn)a - c \equiv b - d \pmod n. Together with compatibility with multiplication, this result shows that the addition of integers modulo n is well defined.

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

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Modular_arithmetic, Basic properties
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Kindstatement, proposition
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UsesCongruence modulo n

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