False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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False: $[a]_{n} [b]_{n} = [0]_{n}$ always implies $[a]_{n} = [0]_{n}$ or $[b]_{n} = [0]_{n}$

Statement

False claim: for every natural modulus $n$ ,

$[a]_{n} [b]_{n} = [0]_{n} ⟹ [a]_{n} = [0]_{n} or [b]_{n} = [0]_{n} .$

Facts & Assumptions

Given: The quotient $Z / 6$ and the classes $[2]_{6}, [3]_{6}$ .

[L1]

Products in $Z / n$ are computed by multiplying representatives (Addition and multiplication on $Z / n$ by $[a]_{n} + [b]_{n} = [a + b]_{n}$ and $[a]_{n} [b]_{n} = [a b]_{n}$ ).

[L2]

Composite moduli admit nonzero classes with zero product; moreover $[a]_{n} = [b]_{n}$ exactly when $a \equiv b (m o d n)$ , which means $n ∣ (a - b)$ (A modulus $n > 1$ is composite exactly when two nonzero classes in $Z / n$ have product $[0]_{n}$ , The congruence class $[a]_{n}$ and the quotient set $Z / n$ , Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

Refutation

technique · direct

1.1

In $Z / 6$ , $[2]_{6} [3]_{6} = [6]_{6} = [0]_{6}$ by [L1].

1.2

Neither $6 ∣ 2$ nor $6 ∣ 3$ , so $[2]_{6} \neq [0]_{6}$ and $[3]_{6} \neq [0]_{6}$ .

2.1

Thus the antecedent of the claimed implication holds while both alternatives in its conclusion fail, so the claim is false.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Conrad, Modular Arithmetic (standard reference, not scraped)
University of Colorado, proof practice on zero divisors (standard reference, not scraped)

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