OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 33815: Normalize and Qualify a Unit before Defining Multiplicative Order

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 170

← Question derivation index · Original question batch

R0066-33815 · q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0015

Original question and givens

Use exact modulus n=17 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

  • G001 — Use exact modulus n=17 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-mod47-morder-qualification, [a, domain, n, operation]]

After

derived = [morder-qualification, Exact_multiplicative_order_mod_n]

Conditions: none

THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.

Step identifiers and dependencies

Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001

S002 · implies

Before

source roles = [ORIGINAL representative, 2, 17]

After

derived = [0, 2]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S001

S003 · implies

Before

source roles = [17, 2, 2, 1]

After

derived = [[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 16]

Conditions: none

FIRST require n≥2,normalize into0..n−1 and qualify gcd1.Enumerate ALL unit residues to obtain phi(n);do not replace general phi with n−1 or assign a numeric order to a nonunit.

Step identifiers and dependencies

Rule: Order_ALL_residues_UNIT_gcd_and_complete_phi_carrier · Uses: G001, S002

S004 · calculation

Before

Order ORIGINAL previous residue times unit exponent1 := mul(1, 2)

After

Order ORIGINAL previous residue times unit exponent1 = 2

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [first-return exponent1, 2, 17]

After

derived = [0, 2]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S004

S006 · calculation

Before

Order ORIGINAL previous residue times unit exponent2 := mul(2, 2)

After

Order ORIGINAL previous residue times unit exponent2 = 4

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [first-return exponent2, 4, 17]

After

derived = [0, 4]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S006

S008 · calculation

Before

Order ORIGINAL previous residue times unit exponent3 := mul(4, 2)

After

Order ORIGINAL previous residue times unit exponent3 = 8

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [first-return exponent3, 8, 17]

After

derived = [0, 8]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S008

S010 · calculation

Before

Order ORIGINAL previous residue times unit exponent4 := mul(8, 2)

After

Order ORIGINAL previous residue times unit exponent4 = 16

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [first-return exponent4, 16, 17]

After

derived = [0, 16]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S010

S012 · calculation

Before

Order ORIGINAL previous residue times unit exponent5 := mul(16, 2)

After

Order ORIGINAL previous residue times unit exponent5 = 32

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

source roles = [first-return exponent5, 32, 17]

After

derived = [1, 15]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S012

S014 · calculation

Before

Order ORIGINAL previous residue times unit exponent6 := mul(15, 2)

After

Order ORIGINAL previous residue times unit exponent6 = 30

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

source roles = [first-return exponent6, 30, 17]

After

derived = [1, 13]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S014

S016 · calculation

Before

Order ORIGINAL previous residue times unit exponent7 := mul(13, 2)

After

Order ORIGINAL previous residue times unit exponent7 = 26

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [first-return exponent7, 26, 17]

After

derived = [1, 9]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S016

S018 · calculation

Before

Order ORIGINAL previous residue times unit exponent8 := mul(9, 2)

After

Order ORIGINAL previous residue times unit exponent8 = 18

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

source roles = [first-return exponent8, 18, 17]

After

derived = [1, 1]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S018

S020 · implies

Before

source roles = [17, 2, 16]

After

derived = [[[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], 8]

Conditions: none

The SOURCE computes the complete first-return prefix EVEN when qualification outputs only gcd/phi.Retain this UNUSED computation obligation.Start at POSITIVE exponent1 with no earlier return.Do not claim order=phi,a primitive root or additive order.

Step identifiers and dependencies

Rule: Order_SOURCE_UNUSED_qualification_return_table_still_checked · Uses: G001, S019

Reverse checks

  • S020 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S019 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S018 · premise_and_result_audit — Recompute Order ORIGINAL previous residue times unit exponent8 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S016 · premise_and_result_audit — Recompute Order ORIGINAL previous residue times unit exponent7 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S014 · premise_and_result_audit — Recompute Order ORIGINAL previous residue times unit exponent6 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S012 · premise_and_result_audit — Recompute Order ORIGINAL previous residue times unit exponent5 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S010 · premise_and_result_audit — Recompute Order ORIGINAL previous residue times unit exponent4 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S008 · premise_and_result_audit — Recompute Order ORIGINAL previous residue times unit exponent3 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S006 · premise_and_result_audit — Recompute Order ORIGINAL previous residue times unit exponent2 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S004 · premise_and_result_audit — Recompute Order ORIGINAL previous residue times unit exponent1 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S002 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.

Result

(17, 2, 2, 1, 16, Yes, unit_mod_n_qualified)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: ff9a876701a8a1dfab91fc3913b028a0ea17ab4b8416faf54a78531d07c34aef
  • candidate record SHA-256: a960d05bbda6b64c7ec9f1e737dd51a905018e4c3c8b1858070c25f3f3a09894
  • semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
  • question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON