Step-by-step derivation
S001 · implies
Before
selected modulus representative1:original nonnegative Euclidean inputs=[1, 9]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 9, 0, 1], [9, 1, 9, 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 · implies
Before
selected modulus:original representative k=1 in1..9;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, S001
S003 · implies
Before
selected modulus representative2:original nonnegative Euclidean inputs=[2, 9]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 9, 0, 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, S002
S004 · implies
Before
selected modulus:original representative k=2 in1..9;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
selected modulus representative3:original nonnegative Euclidean inputs=[3, 9]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 9, 0, 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, S004
S006 · implies
Before
selected modulus:original representative k=3 in1..9;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, S005
S007 · implies
Before
selected modulus representative4:original nonnegative Euclidean inputs=[4, 9]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 9, 0, 4], [9, 4, 2, 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, S006
S008 · implies
Before
selected modulus:original representative k=4 in1..9;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, S007
S009 · implies
Before
selected modulus representative5:original nonnegative Euclidean inputs=[5, 9]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 9, 0, 5], [9, 5, 1, 4], [5, 4, 1, 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, S008
S010 · implies
Before
selected modulus:original representative k=5 in1..9;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
selected modulus representative6:original nonnegative Euclidean inputs=[6, 9]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 9, 0, 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, S010
S012 · implies
Before
selected modulus:original representative k=6 in1..9;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, S011
S013 · implies
Before
selected modulus representative7:original nonnegative Euclidean inputs=[7, 9]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 9, 0, 7], [9, 7, 1, 2], [7, 2, 3, 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, S012
S014 · implies
Before
selected modulus:original representative k=7 in1..9;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, S013
S015 · implies
Before
selected modulus representative8:original nonnegative Euclidean inputs=[8, 9]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 9, 0, 8], [9, 8, 1, 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, S014
S016 · implies
Before
selected modulus:original representative k=8 in1..9;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
selected modulus representative9:original nonnegative Euclidean inputs=[9, 9]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 9, 1, 0]];terminal positive gcd=9
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
selected modulus:original representative k=9 in1..9;verified gcd(k,n)=9
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, S017
S019 · calculation
Before
selected modulus exact coprime indicator count := sum(1, 1, 0, 1, 1, 0, 1, 1, 0)
After
selected modulus exact coprime indicator count = 6
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, S018
S020 · implies
Before
selected modulus:complete representative range1..9;all selected coprime representatives=[1, 2, 4, 5, 7, 8]
After
phi(9)=6;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, S019
S021 · implies
Before
original n=9;exact phi=6;first proper trial divisor up to floor(sqrt(n))=3
After
n=1 handled before primality;classification=composite_no_prime_shortcut
Conditions: none
One is not prime but phi(1)=1.For n≥2,no divisor up to its square root means primality;then every representative1 through n−1 is coprime and phi=n−1.Composite inputs retain the complete count rather than a prime-only shortcut.
Step identifiers and dependencies
Rule: totient_one_prime_composite_distinct_cases · Uses: G001, S020