Original question and givens
Use exact modulus n=20 and integer a=3. 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=20 and integer a=3. 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).
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, 3, 20]
After
derived = [0, 3]
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 = [20, 3, 3, 1]
After
derived = [[1, 3, 7, 9, 11, 13, 17, 19], 8]
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, 3)
After
Order ORIGINAL previous residue times unit exponent1 = 3
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, 3, 20]
After
derived = [0, 3]
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(3, 3)
After
Order ORIGINAL previous residue times unit exponent2 = 9
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, 9, 20]
After
derived = [0, 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, S006
S008 · calculation
Before
Order ORIGINAL previous residue times unit exponent3 := mul(9, 3)
After
Order ORIGINAL previous residue times unit exponent3 = 27
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, 27, 20]
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 dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S008
S010 · calculation
Before
Order ORIGINAL previous residue times unit exponent4 := mul(7, 3)
After
Order ORIGINAL previous residue times unit exponent4 = 21
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, 21, 20]
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, S010
S012 · implies
Before
source roles = [20, 3, 8]
After
derived = [[[1, 3], [2, 9], [3, 7], [4, 1]], 4]
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, S011