Alphabeta Math

Congruences and Modular Arithmetic: Examples and Counterexamples

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✓ 10 results · all verified · 1 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

The system x≡2 (mod 3), x≡3 (mod 5), x≡2 (mod 7) has the unique solution x≡23 (mod 105)

Example

The congruence 14x≡30 (mod 100) has exactly the two solutions x≡45,95 (mod 100)

Example

(ℤ/12)^×={[1],[5],[7],[11]} and φ(12)=4

Example

The Chinese remainder bijection ℤ/12→ℤ/3×ℤ/4 computed on all standard representatives

Example

(ℤ/8)^×={[1],[3],[5],[7]} is not cyclic because every element squares to [1]

Example

φ(360)=96 by both inclusion-exclusion and the prime-power product formula

Counterexample

The system x≡1 (mod 2), x≡2 (mod 4) has no solution, so coprimality in the Chinese remainder theorem cannot be dropped

Example

A modulus n>1 is composite exactly when two nonzero classes in ℤ/n have product [0]ₙ

False statement

False: [a]ₙ[b]ₙ=[0]ₙ always implies [a]ₙ=[0]ₙ or [b]ₙ=[0]ₙ

Counterexample

ℤ/1 has one element and satisfies [0]₁=[1]₁, so it is not a field

Sources