Factor Lists and Common Divisors

Formula and method

List positive divisors in ascending order. For two integers, keep the divisors they share. A negative integer has the same positive divisors as its absolute value; including signs doubles the finite list.

d is a positive common factor of a and b when d divides both integers exactly.

Worked example

Enter these known values and leave the other values blank.

Assumptions and limitations

• Enter one or two integers with magnitude at most10000000000. A blank second input is omitted; it is not zero.
• Positive lists are ascending. For a single negative input, the signed list alternates each positive divisor and its negative, preserving the source presentation.
• Zero has infinitely many nonzero divisors and no finite list/count. A zero paired with a nonzero integer shares that nonzero integer’s divisors. Two zeros have infinitely many common nonzero divisors.
• This verified computational domain bounds each trial loop to100000 and each signed list to65536characters. Larger magnitudes are explicitly rejected rather than creating an unbounded browser workload.
• The source only displays each distinct operand’s list once when both inputs are equal. Counts and common-factor summaries are still computed consistently.
• 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

• Captured factor and common-factor controls and list ordering

Translate the captured factor/count helpers and callback lists into bounded exact BigInt divisor enumeration, retaining both optional integer inputs, ascending positive lists, alternating positive/negative lists, individual counts and common-factor count/list. Keep zero mathematically unbounded without inventing a finite zero list. Original JavaScript is retained as immutable research evidence and is not executed in product. Bind five numeric summaries and five declared plain-text list exports to the complete captured common-factor source and exact compiled schema. Accept integer magnitudes at most10000000000 under a proved100000-trial operand budget and maximum2304positive divisors,65536characters per signed list and250000characters overall. Zero has no finite count/list; one zero paired with a nonzero integer has the other integer’s finite common factors. Preserve optional-input operation in both UI and API under this literal source only. Plaintext is rebuilt from current valid operands and cached for at most eight operand pairs; no raw HTML, string expression evaluation or arbitrary exported names are allowed.

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• Euclidean Residues and Signed Remainder Conventions
• Floor: Round Down to an Integer
• Imaginary Unit Integer Power Cycle