Step-by-step derivation
S001 · implies
Before
totient multiplicativity coprimality premise:original nonnegative Euclidean inputs=[2, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 21, 0, 2], [21, 2, 10, 1], [2, 1, 2, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001
S002 · calculation
Before
totient multiplicativity original modulus product := mul(2, 21)
After
totient multiplicativity original modulus product = 42
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · implies
Before
left modulus representative1:original nonnegative Euclidean inputs=[1, 2]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 2, 0, 1], [2, 1, 2, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S002
S004 · implies
Before
left modulus:original representative k=1 in1..2;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S003
S005 · implies
Before
left modulus representative2:original nonnegative Euclidean inputs=[2, 2]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 2, 1, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S004
S006 · implies
Before
left modulus:original representative k=2 in1..2;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S005
S007 · calculation
Before
left modulus exact coprime indicator count := sum(1, 0)
After
left modulus exact coprime indicator count = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
left modulus:complete representative range1..2;all selected coprime representatives=[1]
After
phi(2)=1;representatives_1_through_n
Conditions: none
Sum every indicator for the exact count.This is complete finite counting for the supplied n,not a general formula extrapolated from samples.
Step identifiers and dependencies
Rule: totient_complete_finite_count_not_sample_formula · Uses: G001, S007
S009 · implies
Before
right modulus representative1:original nonnegative Euclidean inputs=[1, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 21, 0, 1], [21, 1, 21, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S008
S010 · implies
Before
right modulus:original representative k=1 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S009
S011 · implies
Before
right modulus representative2:original nonnegative Euclidean inputs=[2, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 21, 0, 2], [21, 2, 10, 1], [2, 1, 2, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S010
S012 · implies
Before
right modulus:original representative k=2 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S011
S013 · implies
Before
right modulus representative3:original nonnegative Euclidean inputs=[3, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 21, 0, 3], [21, 3, 7, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S012
S014 · implies
Before
right modulus:original representative k=3 in1..21;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S013
S015 · implies
Before
right modulus representative4:original nonnegative Euclidean inputs=[4, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 21, 0, 4], [21, 4, 5, 1], [4, 1, 4, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S014
S016 · implies
Before
right modulus:original representative k=4 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S015
S017 · implies
Before
right modulus representative5:original nonnegative Euclidean inputs=[5, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 21, 0, 5], [21, 5, 4, 1], [5, 1, 5, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S016
S018 · implies
Before
right modulus:original representative k=5 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S017
S019 · implies
Before
right modulus representative6:original nonnegative Euclidean inputs=[6, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 21, 0, 6], [21, 6, 3, 3], [6, 3, 2, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S018
S020 · implies
Before
right modulus:original representative k=6 in1..21;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S019
S021 · implies
Before
right modulus representative7:original nonnegative Euclidean inputs=[7, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 21, 0, 7], [21, 7, 3, 0]];terminal positive gcd=7
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S020
S022 · implies
Before
right modulus:original representative k=7 in1..21;verified gcd(k,n)=7
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S021
S023 · implies
Before
right modulus representative8:original nonnegative Euclidean inputs=[8, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 21, 0, 8], [21, 8, 2, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S022
S024 · implies
Before
right modulus:original representative k=8 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S023
S025 · implies
Before
right modulus representative9:original nonnegative Euclidean inputs=[9, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 21, 0, 9], [21, 9, 2, 3], [9, 3, 3, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S024
S026 · implies
Before
right modulus:original representative k=9 in1..21;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S025
S027 · implies
Before
right modulus representative10:original nonnegative Euclidean inputs=[10, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 21, 0, 10], [21, 10, 2, 1], [10, 1, 10, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S026
S028 · implies
Before
right modulus:original representative k=10 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S027
S029 · implies
Before
right modulus representative11:original nonnegative Euclidean inputs=[11, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 21, 0, 11], [21, 11, 1, 10], [11, 10, 1, 1], [10, 1, 10, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S028
S030 · implies
Before
right modulus:original representative k=11 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S029
S031 · implies
Before
right modulus representative12:original nonnegative Euclidean inputs=[12, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 21, 0, 12], [21, 12, 1, 9], [12, 9, 1, 3], [9, 3, 3, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S030
S032 · implies
Before
right modulus:original representative k=12 in1..21;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S031
S033 · implies
Before
right modulus representative13:original nonnegative Euclidean inputs=[13, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 21, 0, 13], [21, 13, 1, 8], [13, 8, 1, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S032
S034 · implies
Before
right modulus:original representative k=13 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S033
S035 · implies
Before
right modulus representative14:original nonnegative Euclidean inputs=[14, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 21, 0, 14], [21, 14, 1, 7], [14, 7, 2, 0]];terminal positive gcd=7
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S034
S036 · implies
Before
right modulus:original representative k=14 in1..21;verified gcd(k,n)=7
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S035
S037 · implies
Before
right modulus representative15:original nonnegative Euclidean inputs=[15, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 21, 0, 15], [21, 15, 1, 6], [15, 6, 2, 3], [6, 3, 2, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S036
S038 · implies
Before
right modulus:original representative k=15 in1..21;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S037
S039 · implies
Before
right modulus representative16:original nonnegative Euclidean inputs=[16, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 21, 0, 16], [21, 16, 1, 5], [16, 5, 3, 1], [5, 1, 5, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S038
S040 · implies
Before
right modulus:original representative k=16 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S039
S041 · implies
Before
right modulus representative17:original nonnegative Euclidean inputs=[17, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 21, 0, 17], [21, 17, 1, 4], [17, 4, 4, 1], [4, 1, 4, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S040
S042 · implies
Before
right modulus:original representative k=17 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S041
S043 · implies
Before
right modulus representative18:original nonnegative Euclidean inputs=[18, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 21, 0, 18], [21, 18, 1, 3], [18, 3, 6, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S042
S044 · implies
Before
right modulus:original representative k=18 in1..21;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S043
S045 · implies
Before
right modulus representative19:original nonnegative Euclidean inputs=[19, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 21, 0, 19], [21, 19, 1, 2], [19, 2, 9, 1], [2, 1, 2, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S044
S046 · implies
Before
right modulus:original representative k=19 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S045
S047 · implies
Before
right modulus representative20:original nonnegative Euclidean inputs=[20, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 21, 0, 20], [21, 20, 1, 1], [20, 1, 20, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S046
S048 · implies
Before
right modulus:original representative k=20 in1..21;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S047
S049 · implies
Before
right modulus representative21:original nonnegative Euclidean inputs=[21, 21]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 21, 1, 0]];terminal positive gcd=21
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S048
S050 · implies
Before
right modulus:original representative k=21 in1..21;verified gcd(k,n)=21
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S049
S051 · calculation
Before
right modulus exact coprime indicator count := sum(1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0)
After
right modulus exact coprime indicator count = 12
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S050
S052 · implies
Before
right modulus:complete representative range1..21;all selected coprime representatives=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]
After
phi(21)=12;representatives_1_through_n
Conditions: none
Sum every indicator for the exact count.This is complete finite counting for the supplied n,not a general formula extrapolated from samples.
Step identifiers and dependencies
Rule: totient_complete_finite_count_not_sample_formula · Uses: G001, S051
S053 · implies
Before
product modulus representative1:original nonnegative Euclidean inputs=[1, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 42, 0, 1], [42, 1, 42, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S052
S054 · implies
Before
product modulus:original representative k=1 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S053
S055 · implies
Before
product modulus representative2:original nonnegative Euclidean inputs=[2, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 42, 0, 2], [42, 2, 21, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S054
S056 · implies
Before
product modulus:original representative k=2 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S055
S057 · implies
Before
product modulus representative3:original nonnegative Euclidean inputs=[3, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 42, 0, 3], [42, 3, 14, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S056
S058 · implies
Before
product modulus:original representative k=3 in1..42;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S057
S059 · implies
Before
product modulus representative4:original nonnegative Euclidean inputs=[4, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 42, 0, 4], [42, 4, 10, 2], [4, 2, 2, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S058
S060 · implies
Before
product modulus:original representative k=4 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S059
S061 · implies
Before
product modulus representative5:original nonnegative Euclidean inputs=[5, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 42, 0, 5], [42, 5, 8, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S060
S062 · implies
Before
product modulus:original representative k=5 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S061
S063 · implies
Before
product modulus representative6:original nonnegative Euclidean inputs=[6, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 42, 0, 6], [42, 6, 7, 0]];terminal positive gcd=6
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S062
S064 · implies
Before
product modulus:original representative k=6 in1..42;verified gcd(k,n)=6
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S063
S065 · implies
Before
product modulus representative7:original nonnegative Euclidean inputs=[7, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 42, 0, 7], [42, 7, 6, 0]];terminal positive gcd=7
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S064
S066 · implies
Before
product modulus:original representative k=7 in1..42;verified gcd(k,n)=7
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S065
S067 · implies
Before
product modulus representative8:original nonnegative Euclidean inputs=[8, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 42, 0, 8], [42, 8, 5, 2], [8, 2, 4, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S066
S068 · implies
Before
product modulus:original representative k=8 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S067
S069 · implies
Before
product modulus representative9:original nonnegative Euclidean inputs=[9, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 42, 0, 9], [42, 9, 4, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S068
S070 · implies
Before
product modulus:original representative k=9 in1..42;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S069
S071 · implies
Before
product modulus representative10:original nonnegative Euclidean inputs=[10, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 42, 0, 10], [42, 10, 4, 2], [10, 2, 5, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S070
S072 · implies
Before
product modulus:original representative k=10 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S071
S073 · implies
Before
product modulus representative11:original nonnegative Euclidean inputs=[11, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 42, 0, 11], [42, 11, 3, 9], [11, 9, 1, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S072
S074 · implies
Before
product modulus:original representative k=11 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S073
S075 · implies
Before
product modulus representative12:original nonnegative Euclidean inputs=[12, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 42, 0, 12], [42, 12, 3, 6], [12, 6, 2, 0]];terminal positive gcd=6
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S074
S076 · implies
Before
product modulus:original representative k=12 in1..42;verified gcd(k,n)=6
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S075
S077 · implies
Before
product modulus representative13:original nonnegative Euclidean inputs=[13, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 42, 0, 13], [42, 13, 3, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S076
S078 · implies
Before
product modulus:original representative k=13 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S077
S079 · implies
Before
product modulus representative14:original nonnegative Euclidean inputs=[14, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 42, 0, 14], [42, 14, 3, 0]];terminal positive gcd=14
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S078
S080 · implies
Before
product modulus:original representative k=14 in1..42;verified gcd(k,n)=14
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S079
S081 · implies
Before
product modulus representative15:original nonnegative Euclidean inputs=[15, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 42, 0, 15], [42, 15, 2, 12], [15, 12, 1, 3], [12, 3, 4, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S080
S082 · implies
Before
product modulus:original representative k=15 in1..42;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S081
S083 · implies
Before
product modulus representative16:original nonnegative Euclidean inputs=[16, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 42, 0, 16], [42, 16, 2, 10], [16, 10, 1, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S082
S084 · implies
Before
product modulus:original representative k=16 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S083
S085 · implies
Before
product modulus representative17:original nonnegative Euclidean inputs=[17, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 42, 0, 17], [42, 17, 2, 8], [17, 8, 2, 1], [8, 1, 8, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S084
S086 · implies
Before
product modulus:original representative k=17 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S085
S087 · implies
Before
product modulus representative18:original nonnegative Euclidean inputs=[18, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 42, 0, 18], [42, 18, 2, 6], [18, 6, 3, 0]];terminal positive gcd=6
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S086
S088 · implies
Before
product modulus:original representative k=18 in1..42;verified gcd(k,n)=6
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S087
S089 · implies
Before
product modulus representative19:original nonnegative Euclidean inputs=[19, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 42, 0, 19], [42, 19, 2, 4], [19, 4, 4, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S088
S090 · implies
Before
product modulus:original representative k=19 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S089
S091 · implies
Before
product modulus representative20:original nonnegative Euclidean inputs=[20, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 42, 0, 20], [42, 20, 2, 2], [20, 2, 10, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S090
S092 · implies
Before
product modulus:original representative k=20 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S091
S093 · implies
Before
product modulus representative21:original nonnegative Euclidean inputs=[21, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 42, 0, 21], [42, 21, 2, 0]];terminal positive gcd=21
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S092
S094 · implies
Before
product modulus:original representative k=21 in1..42;verified gcd(k,n)=21
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S093
S095 · implies
Before
product modulus representative22:original nonnegative Euclidean inputs=[22, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 42, 0, 22], [42, 22, 1, 20], [22, 20, 1, 2], [20, 2, 10, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S094
S096 · implies
Before
product modulus:original representative k=22 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S095
S097 · implies
Before
product modulus representative23:original nonnegative Euclidean inputs=[23, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 42, 0, 23], [42, 23, 1, 19], [23, 19, 1, 4], [19, 4, 4, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S096
S098 · implies
Before
product modulus:original representative k=23 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S097
S099 · implies
Before
product modulus representative24:original nonnegative Euclidean inputs=[24, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 42, 0, 24], [42, 24, 1, 18], [24, 18, 1, 6], [18, 6, 3, 0]];terminal positive gcd=6
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S098
S100 · implies
Before
product modulus:original representative k=24 in1..42;verified gcd(k,n)=6
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S099
S101 · implies
Before
product modulus representative25:original nonnegative Euclidean inputs=[25, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 42, 0, 25], [42, 25, 1, 17], [25, 17, 1, 8], [17, 8, 2, 1], [8, 1, 8, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S100
S102 · implies
Before
product modulus:original representative k=25 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S101
S103 · implies
Before
product modulus representative26:original nonnegative Euclidean inputs=[26, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 42, 0, 26], [42, 26, 1, 16], [26, 16, 1, 10], [16, 10, 1, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S102
S104 · implies
Before
product modulus:original representative k=26 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S103
S105 · implies
Before
product modulus representative27:original nonnegative Euclidean inputs=[27, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 42, 0, 27], [42, 27, 1, 15], [27, 15, 1, 12], [15, 12, 1, 3], [12, 3, 4, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S104
S106 · implies
Before
product modulus:original representative k=27 in1..42;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S105
S107 · implies
Before
product modulus representative28:original nonnegative Euclidean inputs=[28, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 42, 0, 28], [42, 28, 1, 14], [28, 14, 2, 0]];terminal positive gcd=14
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S106
S108 · implies
Before
product modulus:original representative k=28 in1..42;verified gcd(k,n)=14
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S107
S109 · implies
Before
product modulus representative29:original nonnegative Euclidean inputs=[29, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 42, 0, 29], [42, 29, 1, 13], [29, 13, 2, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S108
S110 · implies
Before
product modulus:original representative k=29 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S109
S111 · implies
Before
product modulus representative30:original nonnegative Euclidean inputs=[30, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 42, 0, 30], [42, 30, 1, 12], [30, 12, 2, 6], [12, 6, 2, 0]];terminal positive gcd=6
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S110
S112 · implies
Before
product modulus:original representative k=30 in1..42;verified gcd(k,n)=6
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S111
S113 · implies
Before
product modulus representative31:original nonnegative Euclidean inputs=[31, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[31, 42, 0, 31], [42, 31, 1, 11], [31, 11, 2, 9], [11, 9, 1, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S112
S114 · implies
Before
product modulus:original representative k=31 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S113
S115 · implies
Before
product modulus representative32:original nonnegative Euclidean inputs=[32, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 42, 0, 32], [42, 32, 1, 10], [32, 10, 3, 2], [10, 2, 5, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S114
S116 · implies
Before
product modulus:original representative k=32 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S115
S117 · implies
Before
product modulus representative33:original nonnegative Euclidean inputs=[33, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[33, 42, 0, 33], [42, 33, 1, 9], [33, 9, 3, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S116
S118 · implies
Before
product modulus:original representative k=33 in1..42;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S117
S119 · implies
Before
product modulus representative34:original nonnegative Euclidean inputs=[34, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 42, 0, 34], [42, 34, 1, 8], [34, 8, 4, 2], [8, 2, 4, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S118
S120 · implies
Before
product modulus:original representative k=34 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S119
S121 · implies
Before
product modulus representative35:original nonnegative Euclidean inputs=[35, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 42, 0, 35], [42, 35, 1, 7], [35, 7, 5, 0]];terminal positive gcd=7
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S120
S122 · implies
Before
product modulus:original representative k=35 in1..42;verified gcd(k,n)=7
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S121
S123 · implies
Before
product modulus representative36:original nonnegative Euclidean inputs=[36, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[36, 42, 0, 36], [42, 36, 1, 6], [36, 6, 6, 0]];terminal positive gcd=6
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S122
S124 · implies
Before
product modulus:original representative k=36 in1..42;verified gcd(k,n)=6
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S123
S125 · implies
Before
product modulus representative37:original nonnegative Euclidean inputs=[37, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[37, 42, 0, 37], [42, 37, 1, 5], [37, 5, 7, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S124
S126 · implies
Before
product modulus:original representative k=37 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S125
S127 · implies
Before
product modulus representative38:original nonnegative Euclidean inputs=[38, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[38, 42, 0, 38], [42, 38, 1, 4], [38, 4, 9, 2], [4, 2, 2, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S126
S128 · implies
Before
product modulus:original representative k=38 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S127
S129 · implies
Before
product modulus representative39:original nonnegative Euclidean inputs=[39, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[39, 42, 0, 39], [42, 39, 1, 3], [39, 3, 13, 0]];terminal positive gcd=3
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S128
S130 · implies
Before
product modulus:original representative k=39 in1..42;verified gcd(k,n)=3
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S129
S131 · implies
Before
product modulus representative40:original nonnegative Euclidean inputs=[40, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[40, 42, 0, 40], [42, 40, 1, 2], [40, 2, 20, 0]];terminal positive gcd=2
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S130
S132 · implies
Before
product modulus:original representative k=40 in1..42;verified gcd(k,n)=2
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S131
S133 · implies
Before
product modulus representative41:original nonnegative Euclidean inputs=[41, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[41, 42, 0, 41], [42, 41, 1, 1], [41, 1, 41, 0]];terminal positive gcd=1
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S132
S134 · implies
Before
product modulus:original representative k=41 in1..42;verified gcd(k,n)=1
After
coprime indicator=1;include representative=Yes
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S133
S135 · implies
Before
product modulus representative42:original nonnegative Euclidean inputs=[42, 42]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[42, 42, 1, 0]];terminal positive gcd=42
Conditions: none
Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.
Step identifiers and dependencies
Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S134
S136 · implies
Before
product modulus:original representative k=42 in1..42;verified gcd(k,n)=42
After
coprime indicator=0;include representative=No
Conditions: none
Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.
Step identifiers and dependencies
Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S135
S137 · calculation
Before
product modulus exact coprime indicator count := sum(1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0)
After
product modulus exact coprime indicator count = 12
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S136
S138 · implies
Before
product modulus:complete representative range1..42;all selected coprime representatives=[1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41]
After
phi(42)=12;representatives_1_through_n
Conditions: none
Sum every indicator for the exact count.This is complete finite counting for the supplied n,not a general formula extrapolated from samples.
Step identifiers and dependencies
Rule: totient_complete_finite_count_not_sample_formula · Uses: G001, S137
S139 · calculation
Before
totient supplied instance product of counts := mul(1, 12)
After
totient supplied instance product of counts = 12
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S138
S140 · implies
Before
original coprime inputs=[2, 21];gcd=1;product modulus=42;exact totients=[1, 12, 12]
After
phi(m)*phi(n)=12;same supplied-instance equality=Yes;instance_only_with_coprimality
Conditions: none
First verify coprimality and all three allowed moduli,then count the three totients completely and compare.This checks the supplied instance only.Non-coprime inputs do not qualify,and numeric agreement is not a proof of the universal multiplicativity theorem.
Step identifiers and dependencies
Rule: totient_coprime_instance_not_unrestricted_multiplicativity_proof · Uses: G001, S139