--- name: math-thumb-number-theory description: "Apply Chapter 8 (Number Theory: Reduce Large Arithmetic to Small Structure) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to reduce integer calculations while preserving divisibility and invertibility." --- # Number Theory: Reduce Large Arithmetic to Small Structure Use this chapter to help the reader make a checked mathematical decision. All 10 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Integers and modulus; whether an exact answer or scale estimate is needed; coprimality; one-number versus many-number workload. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Do you need a gcd but not a factorization?** Use the Euclidean algorithm. Record its equations if an inverse or Bézout coefficients may be needed next. - **Is a complete factorization already known?** Translate exponent choices directly into a divisor count. Do not enumerate divisors unless the list itself is required. - **Do you need an aggregate prime estimate?** Use $n/\log n$ for scale. If you need exact primes, choose between trial division, a sieve, and a larger-scale primality method based on workload. - **Is the answer requested modulo $m$?** Reduce throughout addition and multiplication. Before dividing, test invertibility with a gcd. - **Can the modulus be separated into coprime factors?** Solve the smaller congruences and reconstruct with the Chinese remainder theorem. - **Is the obstacle a huge exponent?** Verify coprimality, identify a valid group period, and reduce the exponent before powering. - **Is the question about the power of one prime in a product, sum, factorial, or gcd?** Replace full integers with valuations and watch for equal-valuation cancellation. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Check coprimality before modular inversion, exponent-period reduction, or the simplest CRT formula. Prime-count estimates do not certify primality. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 8.1.1, 8.2.2, 8.2.3 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C08-D01: Find an inverse only when arithmetic permits it**: rule 8.2.2. - **C08-D02: Intersect congruences in a small exact search**: rule 8.2.3. - **C08-D03: Compare counted primes with their asymptotic scale**: rule 8.1.3. - **C08-D04: Reduce a huge exponent by the period of the powers**: rule 8.3.3. - **C08-D05: See why trial division can stop at the square root**: rule 8.3.1. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.