# Question 33816: Normalize and Qualify a Unit before Defining Multiplicative Order

[Return to catalogue page 170](https://ocme.evemisslab.com/en/questions/page/170/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations/#q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0016)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0066-33816 · q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0016



## Original question and givens

Use exact modulus n=18 and integer a=5. 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=18 and integer a=5. 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.


- [mko-multiplicative-order-mod-n](https://ocme.evemisslab.com/en/mko/mko-multiplicative-order-mod-n/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001





S002 · impliesBefore

source roles = [ORIGINAL representative, 5, 18]

After

derived = [0, 5]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S001





S003 · impliesBefore

source roles = [18, 5, 5, 1]

After

derived = [[1, 5, 7, 11, 13, 17], 6]

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 dependenciesRule: Order_ALL_residues_UNIT_gcd_and_complete_phi_carrier · Uses: G001, S002





S004 · calculationBefore

Order ORIGINAL previous residue times unit exponent1 := mul(1, 5)

After

Order ORIGINAL previous residue times unit exponent1 = 5

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

source roles = [first-return exponent1, 5, 18]

After

derived = [0, 5]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S004





S006 · calculationBefore

Order ORIGINAL previous residue times unit exponent2 := mul(5, 5)

After

Order ORIGINAL previous residue times unit exponent2 = 25

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

source roles = [first-return exponent2, 25, 18]

After

derived = [1, 7]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S006





S008 · calculationBefore

Order ORIGINAL previous residue times unit exponent3 := mul(7, 5)

After

Order ORIGINAL previous residue times unit exponent3 = 35

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · impliesBefore

source roles = [first-return exponent3, 35, 18]

After

derived = [1, 17]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S008





S010 · calculationBefore

Order ORIGINAL previous residue times unit exponent4 := mul(17, 5)

After

Order ORIGINAL previous residue times unit exponent4 = 85

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · impliesBefore

source roles = [first-return exponent4, 85, 18]

After

derived = [4, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S010





S012 · calculationBefore

Order ORIGINAL previous residue times unit exponent5 := mul(13, 5)

After

Order ORIGINAL previous residue times unit exponent5 = 65

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · impliesBefore

source roles = [first-return exponent5, 65, 18]

After

derived = [3, 11]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S012





S014 · calculationBefore

Order ORIGINAL previous residue times unit exponent6 := mul(11, 5)

After

Order ORIGINAL previous residue times unit exponent6 = 55

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · impliesBefore

source roles = [first-return exponent6, 55, 18]

After

derived = [3, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S014





S016 · impliesBefore

source roles = [18, 5, 6]

After

derived = [[[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], 6]

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 dependenciesRule: Order_SOURCE_UNUSED_qualification_return_table_still_checked · Uses: G001, S015









## Reverse checks


- S016 · 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.
- 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

(18, 5, 5, 1, 6, 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: f3d43d9df4c589dd1457dc65c31ae21c093a8efc002980b9ac9cc2ac3903c96f
- candidate record SHA-256: 53e4652ba34c4c1fac60b22866f97ea8b1f63a4c83498b7f72385f67688db5e2
- semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
- question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0016.json)