Step-by-step derivation
S001 · implies
Before
selected modulus representative1:original nonnegative Euclidean inputs=[1, 4]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 4, 0, 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
S002 · implies
Before
selected modulus:original representative k=1 in1..4;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, 4]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 4, 0, 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, S002
S004 · implies
Before
selected modulus:original representative k=2 in1..4;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, S003
S005 · implies
Before
selected modulus representative3:original nonnegative Euclidean inputs=[3, 4]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 4, 0, 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, S004
S006 · implies
Before
selected modulus:original representative k=3 in1..4;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, S005
S007 · implies
Before
selected modulus representative4:original nonnegative Euclidean inputs=[4, 4]
After
division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 4, 1, 0]];terminal positive gcd=4
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..4;verified gcd(k,n)=4
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, S007
S009 · calculation
Before
selected modulus exact coprime indicator count := sum(1, 0, 1, 0)
After
selected modulus exact coprime indicator count = 2
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, S008
S010 · implies
Before
selected modulus:complete representative range1..4;all selected coprime representatives=[1, 3]
After
phi(4)=2;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, S009