Least Common Multiple with Worked Methods

Formula and method

A common multiple must contain the highest required power of every prime. The least such product is the LCM. The GCF identity gives the same answer and can be applied successively to a longer list.

For integers, LCM(a,b)=|a×b|/GCF(a,b). For finite decimals, scale every magnitude by the same10^s, find the integer LCM, then divide by10^s.

Worked example

Enter these known values and leave the other values blank.

Assumptions and limitations

• Enter2 through15 nonzero finite values, each with at most8 decimal places. Negative signs are ignored; repeated values are allowed. Blank fields are omitted.
• The common decimal scaling must leave every positive scaled integer no greater than10000000000. This is a computational budget; reduce magnitudes or decimal places if the combination exceeds it.
• The source has four methods: None, list of multiples, prime factorization and successive GCF. None hides working without hiding the LCM result.
• Multiples include all preceding terms when fewer than95; otherwise the first10, central10 and last5 are sampled with explicit ellipses. The first common value and three subsequent multiples remain. No allocation follows the magnitude of the LCM.
• Each integer factorization uses at most50000 odd trials, the full result has at most151 digits, and all worked plain text is bounded by262144characters with an eight-entry cache.1 has the empty prime product.
• Zero is outside the captured positive-multiple workflow. Under the usual algebraic convention, LCM(0,n)=0; remove zero to use these positive-multiple work lists.
• The source callback rounded decimal inputs to8 places and later explanation rows reused an earlier product/GCF. This implementation requires exact admitted decimals and shows the actual current numbers at every step.
• Inputs accept decimal or e notation and bounded arithmetic expressions only when each intermediate and final value can be verified exactly as a finite decimal. Repeating fractions, functions, symbols and expressions that lose digits are rejected with guidance. Enter their finite decimal value instead; the calculator-specific magnitude and precision limits still apply.

Common questions

References

• Captured15-input/four-method workflow

Translate the captured15-operand optional callback into exact bounded local decimal normalization, BigInt prime-factor/LCM arithmetic and plain-text worked output. Preserve None, list of multiples, prime factorization/highest powers and every sequential GCF row. Correct the captured repetition of an earlier product/GCF in later rows so every displayed step uses its actual current numbers. Preserve source positions, entered order, signs-ignored context and exact shared-scale explanation; no captured JavaScript, HTML or valueMarkdown is executed. Bind the complete LCM capture/compiler/served schema to2–15 nonzero exact finite decimal operands, each at most8 decimal places and magnitude10000000000. Every positive operand after common decimal scaling must remain at most10000000000; dynamically validate this cohort relationship without discarding any operand. Retain all15 progressive controls and four method choices, bounded multiple samples with explicit ellipses, every prime exponent and all14 current GCF steps. Generate only34 declared plain-text exports plus three small readonly numeric summaries. Maximum50000 odd factor trials per operand,151 result digits,262144text characters and eight immutable cached summaries; reject unsupported domains rather than rounding, wrapping or omitting work.

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