least common multiple
Enable an agent to recognise, compute and verify a least common multiple and judge when it correctly represents a shared divisibility or alignment requirement.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
recalled by Codex without web access - no source was read
Researched by: Codex
Purpose and description
Enable an agent to recognise, compute and verify a least common multiple and judge when it correctly represents a shared divisibility or alignment requirement.
The least common multiple of a nonempty finite collection of positive integers is the smallest positive integer divisible by every integer in that collection.
It can be Validate integer inputs and apply declared sign and zero conventions.; Compute an exact LCM for two or more inputs using GCD reduction or prime-exponent maxima.; Verify both common divisibility and leastness of a proposed result.; Update an existing LCM when an additional integer constraint is introduced.; Construct a least common denominator for specified positive integer denominators.; Determine a shared repeat interval for exact integer periods after checking scale and phase assumptions..
Distinguishing features
For nonzero integer inputs, the result is positive and every input divides it; this separates an LCM from a greatest common divisor, which divides every input.
Every common multiple is divisible by the LCM; merely being a common multiple is insufficient.
For positive inputs, the LCM equals their product only when those inputs are pairwise coprime; shared prime factors generally make the LCM smaller.
Each prime exponent in a nonzero LCM is the maximum exponent appearing among the absolute input values, whereas a GCD uses the minimum.
When an input is zero, a result of zero uses an explicit extension of the definition because no positive integer is divisible by zero.
Scope
+ The least positive integer divisible by every input when all inputs are nonzero
+ Normalisation of negative integer inputs to their absolute values
+ The convention that a collection containing zero has least common multiple zero
+ Exact computation and verification using greatest common divisors or prime exponents
+ Finite collections, repeated inputs and incremental accumulation
+ Use in common denominators and compatible periodic alignment problems
- Greatest common divisor as a separately modelled concept
- General prime factorisation methods beyond their role in computing or verifying an LCM
- Complete fraction arithmetic and fraction simplification
- General scheduling with offsets, constraints or uncertain durations
- Least common multiples of polynomials or elements of other algebraic structures
- Extensions to rational inputs, empty collections or infinite collections
Characteristics
- Input integers
- Finite, nonempty collection of integers An LCM is relative to its inputs; its numeric value alone does not identify the divisibility problem.
- Input domain
- Positive integers; nonzero signed integers; integers including zero Determines whether sign normalisation or an explicit zero convention is required.
- Zero convention
- Zero inputs excluded; LCM defined as zero whenever an input is zero Prevents an implementation convention from being confused with the least-positive-common-multiple definition.
- LCM value
- Exact nonnegative integer; dimensionless Records the candidate or verified result without rounding or machine overflow.
- Verification state
- Uncomputed; candidate; common-multiple condition verified; leastness verified; rejected Distinguishes finding any common multiple from establishing the least common multiple.
- Computation method
- GCD reduction; prime-exponent maxima; bounded enumeration; other documented exact method Makes the result reproducible and exposes method-specific failure conditions.
- Application scale
- Optional mapping from integer inputs to counts of a shared exact unit Allows an integer LCM to represent a duration or length only when all inputs use a justified common scale.
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 5 bundles · 9 layers · 15 findings · 23 questions.
Definition and divisibility Establishes what makes a candidate the least common multiple of the supplied integers.
Common divisibility alone does not establish leastness, and zero requires a distinct definitional convention.
Common multiple condition
Checks the direction and completeness of the required divisibility relations.
Every input divides the result
A common multiple must be an integer multiple of each input; reversing this relation tests for a common divisor instead.
- For every input a, is there an integer k such that the candidate m equals a times k? definition
- Have all supplied inputs been checked, including any zero input through the divisibility definition rather than a remainder operation? measurement
Leastness and zero
Separates the least positive common multiple from other common multiples and records the zero extension.
Leastness certificate
For nonzero inputs, the LCM is the smallest positive common multiple and divides every common multiple. Under the zero extension, a collection containing zero has LCM zero.
- What establishes that every common multiple of these inputs is divisible by the proposed LCM? definition
- If an input is zero, is the zero extension declared instead of claiming that zero is a positive common multiple? boundary
Input domain and normalisation Identifies admissible collections and transformations that preserve their LCM.
Signs, zero, repeated values and unsupported domains can change how a request must be interpreted before computation.
Integer domain
Ensures the request belongs to the finite integer interpretation owned by this model.
Admissible input collection
The owned domain is a finite, nonempty integer collection. Negative values can be replaced by absolute values; zero requires the selected convention.
- Is the collection finite and nonempty, and is every value an exact integer? boundary
- Which sign normalisation and zero policy govern these inputs? definition
Redundant inputs
Recognises input constraints that do not alter the result.
Divisibility-preserving reduction
Order and duplicate inputs do not affect the LCM. Among positive inputs, an input that divides another imposes no additional constraint.
- Which positive inputs divide another supplied input and can be removed without changing the LCM? measurement
- Can duplicates and redundant divisors be removed while retaining the original collection for explanation? action
Exact computation and verification Connects executable calculation methods to evidence that the result is correct.
An agent must distinguish an exact verified result from a plausible value produced by rounding, overflow or incomplete factorisation.
GCD reduction
Computes pairwise LCMs and extends them across a finite collection.
GCD-based accumulation
For nonzero a and b, lcm(a,b) equals abs((a/gcd(a,b)) times b). Repeated pairwise application handles a finite collection; a zero input is handled explicitly under the zero extension.
- Does the computation use exact division by the GCD before multiplication and handle zero before any undefined division? action
- Can the numeric representation hold every intermediate value and the final LCM exactly? measurement
Prime-exponent verification
Uses prime factors to explain both common divisibility and leastness for nonzero inputs.
Maximum prime exponents
For each prime appearing in the absolute inputs, the LCM contains exactly the largest input exponent of that prime. The LCM of inputs all equal to one is one.
- Does the candidate contain every required prime with exactly its maximum input exponent and no additional prime factors? measurement
- What calculation record or independently checked factorisation supports the claimed exponent maxima? provenance
Application fit Determines when a divisibility result answers the practical question being asked.
Correct LCM arithmetic can still give the wrong answer when denominator representation, period units or starting phases are misunderstood.
Common denominators
Applies the LCM to specified positive integer denominators while distinguishing representation choices.
Denominator selection
The LCM of specified denominators is their least common multiple. Reducing fractions first can produce a smaller common denominator sufficient to represent the same rational values.
- Is the task to find the LCM of the written denominators or the smallest common denominator after reducing the fractions? boundary
- Are all denominators positive integers, and what exact multiplier converts each selected denominator to the computed LCM? action
Periodic alignment
Applies integer LCMs to exact repeat intervals with compatible starting conditions.
Period and phase conditions
For processes aligned at a shared origin with positive integer periods in a common unit, the LCM gives the first positive return to joint alignment. Different phases require additional congruence checks.
- Are all periods exact positive integer counts of the same unit, and were the processes aligned at the stated origin? boundary
- If starting phases differ, must simultaneous congruences be solved before an alignment time can be asserted? action
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- The sense described is the standard integer-arithmetic concept; polynomial least common multiples are an established extension.
- When extending to arbitrary integers, a common convention uses absolute values and sets lcm(a, 0) = 0, including lcm(0, 0); this requires extending the smallest-positive-common-multiple definition.
- The arithmetic definition is settled; conventions for zero inputs, empty collections, and algebraic normalization should be checked for the intended implementation.
- Which of these check these first hold for the sense of least common multiple this model covers, and on what evidence? provenance
Kinds and varieties
Recalled without web access and unsourced; every item is a lead to verify.
- Least common multiple of two positive integers
- Least common multiple of a finite collection of positive integers
- Polynomial least common multiple, defined up to multiplication by a unit and commonly normalized to be monic over a field
- Which of these kinds and varieties hold for the sense of least common multiple this model covers, and on what evidence? provenance
Identifiers and schemes
Recalled without web access and unsourced; every item is a lead to verify.
- Mathematical notation - lcm(a, b) or LCM(a, b) - Conventional notation for the operation, rather than a registry identifier.
- Which of these identifiers and schemes hold for the sense of least common multiple this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Finding the least common denominator when adding or comparing fractions.
- Determining when periodic events with commensurable periods and a shared starting point next coincide.
- Finding the smallest quantity divisible into each of several specified integer group sizes.
- Computing the order of a permutation as the least common multiple of its disjoint cycle lengths.
- Which of these real-world use hold for the sense of least common multiple this model covers, and on what evidence? provenance
Typical measurements
Recalled without web access and unsourced; every item is a lead to verify.
- Least common multiple of positive integers a and b - No universal typical range; max(a, b) ≤ lcm(a, b) ≤ ab. - Dimensionless
- Which of these typical measurements hold for the sense of least common multiple this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Confusing the least common multiple with the greatest common divisor.
- Using the product of the inputs as their least common multiple when the inputs are not coprime.
- Taking minimum rather than maximum prime-factor exponents when constructing the least common multiple.
- Overflowing fixed-width integer arithmetic when computing ab/gcd(a, b); dividing before multiplying reduces intermediate overflow risk.
- Applying the shared-start recurrence rule to events with different phase offsets, whose coincidence requires solving additional congruence conditions.
- Which of these failure modes and hazards hold for the sense of least common multiple this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- greatest common divisor - The greatest common divisor divides every input; the least common multiple is divisible by every input.
- common multiple - A common multiple is divisible by every input but need not be the smallest positive one.
- least common denominator - The least common denominator applies the least common multiple specifically to the denominators in a collection of fractions.
- product - For two positive integers, their product equals their least common multiple exactly when they are coprime.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of least common multiple this model covers, and on what evidence? provenance
What the second pass must settle
- Which authoritative reference should anchor the registry definition, given that none is recorded?
- Should the published model accept zero inputs by default or require an explicit declaration of the zero extension?
- Does the registry intend this entry to cover algebraic generalisations, or should those be linked as separate neighbouring concepts?
- Which consuming applications require exact unit conversion and phase information alongside the integer LCM?
- What verification evidence should consuming agents retain: reproducible GCD steps, prime-exponent certificates or another exact proof?