Question 16667: Totient Multiplicativity Requires the Coprimality Premise

Original question and givens

Positive integers m=2, n=35 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).

• G001 — Positive integers m=2, n=35 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-euler-totient-function

Step-by-step derivation

totient multiplicativity coprimality premise:original nonnegative Euclidean inputs=[2, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 35, 0, 2], [35, 2, 17, 1], [2, 1, 2, 0]];terminal positive gcd=1

totient multiplicativity original modulus product := mul(2, 35)

totient multiplicativity original modulus product = 70

left modulus representative1:original nonnegative Euclidean inputs=[1, 2]

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

left modulus:original representative k=1 in1..2;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

left modulus representative2:original nonnegative Euclidean inputs=[2, 2]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 2, 1, 0]];terminal positive gcd=2

left modulus:original representative k=2 in1..2;verified gcd(k,n)=2

coprime indicator=0;include representative=No

left modulus exact coprime indicator count := sum(1, 0)

left modulus exact coprime indicator count = 1

left modulus:complete representative range1..2;all selected coprime representatives=[1]

phi(2)=1;representatives_1_through_n

right modulus representative1:original nonnegative Euclidean inputs=[1, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 35, 0, 1], [35, 1, 35, 0]];terminal positive gcd=1

right modulus:original representative k=1 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative2:original nonnegative Euclidean inputs=[2, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 35, 0, 2], [35, 2, 17, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=2 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative3:original nonnegative Euclidean inputs=[3, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 35, 0, 3], [35, 3, 11, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=3 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative4:original nonnegative Euclidean inputs=[4, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 35, 0, 4], [35, 4, 8, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=4 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative5:original nonnegative Euclidean inputs=[5, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 35, 0, 5], [35, 5, 7, 0]];terminal positive gcd=5

right modulus:original representative k=5 in1..35;verified gcd(k,n)=5

coprime indicator=0;include representative=No

right modulus representative6:original nonnegative Euclidean inputs=[6, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 35, 0, 6], [35, 6, 5, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1

right modulus:original representative k=6 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative7:original nonnegative Euclidean inputs=[7, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 35, 0, 7], [35, 7, 5, 0]];terminal positive gcd=7

right modulus:original representative k=7 in1..35;verified gcd(k,n)=7

coprime indicator=0;include representative=No

right modulus representative8:original nonnegative Euclidean inputs=[8, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 35, 0, 8], [35, 8, 4, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=8 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative9:original nonnegative Euclidean inputs=[9, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 35, 0, 9], [35, 9, 3, 8], [9, 8, 1, 1], [8, 1, 8, 0]];terminal positive gcd=1

right modulus:original representative k=9 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative10:original nonnegative Euclidean inputs=[10, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 35, 0, 10], [35, 10, 3, 5], [10, 5, 2, 0]];terminal positive gcd=5

right modulus:original representative k=10 in1..35;verified gcd(k,n)=5

coprime indicator=0;include representative=No

right modulus representative11:original nonnegative Euclidean inputs=[11, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 35, 0, 11], [35, 11, 3, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=11 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative12:original nonnegative Euclidean inputs=[12, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 35, 0, 12], [35, 12, 2, 11], [12, 11, 1, 1], [11, 1, 11, 0]];terminal positive gcd=1

right modulus:original representative k=12 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative13:original nonnegative Euclidean inputs=[13, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 35, 0, 13], [35, 13, 2, 9], [13, 9, 1, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal positive gcd=1

right modulus:original representative k=13 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative14:original nonnegative Euclidean inputs=[14, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 35, 0, 14], [35, 14, 2, 7], [14, 7, 2, 0]];terminal positive gcd=7

right modulus:original representative k=14 in1..35;verified gcd(k,n)=7

coprime indicator=0;include representative=No

right modulus representative15:original nonnegative Euclidean inputs=[15, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 35, 0, 15], [35, 15, 2, 5], [15, 5, 3, 0]];terminal positive gcd=5

right modulus:original representative k=15 in1..35;verified gcd(k,n)=5

coprime indicator=0;include representative=No

right modulus representative16:original nonnegative Euclidean inputs=[16, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 35, 0, 16], [35, 16, 2, 3], [16, 3, 5, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=16 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative17:original nonnegative Euclidean inputs=[17, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 35, 0, 17], [35, 17, 2, 1], [17, 1, 17, 0]];terminal positive gcd=1

right modulus:original representative k=17 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative18:original nonnegative Euclidean inputs=[18, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 35, 0, 18], [35, 18, 1, 17], [18, 17, 1, 1], [17, 1, 17, 0]];terminal positive gcd=1

right modulus:original representative k=18 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative19:original nonnegative Euclidean inputs=[19, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 35, 0, 19], [35, 19, 1, 16], [19, 16, 1, 3], [16, 3, 5, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=19 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative20:original nonnegative Euclidean inputs=[20, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 35, 0, 20], [35, 20, 1, 15], [20, 15, 1, 5], [15, 5, 3, 0]];terminal positive gcd=5

right modulus:original representative k=20 in1..35;verified gcd(k,n)=5

coprime indicator=0;include representative=No

right modulus representative21:original nonnegative Euclidean inputs=[21, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 35, 0, 21], [35, 21, 1, 14], [21, 14, 1, 7], [14, 7, 2, 0]];terminal positive gcd=7

right modulus:original representative k=21 in1..35;verified gcd(k,n)=7

coprime indicator=0;include representative=No

right modulus representative22:original nonnegative Euclidean inputs=[22, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 35, 0, 22], [35, 22, 1, 13], [22, 13, 1, 9], [13, 9, 1, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal positive gcd=1

right modulus:original representative k=22 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative23:original nonnegative Euclidean inputs=[23, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 35, 0, 23], [35, 23, 1, 12], [23, 12, 1, 11], [12, 11, 1, 1], [11, 1, 11, 0]];terminal positive gcd=1

right modulus:original representative k=23 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative24:original nonnegative Euclidean inputs=[24, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 35, 0, 24], [35, 24, 1, 11], [24, 11, 2, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=24 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative25:original nonnegative Euclidean inputs=[25, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 35, 0, 25], [35, 25, 1, 10], [25, 10, 2, 5], [10, 5, 2, 0]];terminal positive gcd=5

right modulus:original representative k=25 in1..35;verified gcd(k,n)=5

coprime indicator=0;include representative=No

right modulus representative26:original nonnegative Euclidean inputs=[26, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 35, 0, 26], [35, 26, 1, 9], [26, 9, 2, 8], [9, 8, 1, 1], [8, 1, 8, 0]];terminal positive gcd=1

right modulus:original representative k=26 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative27:original nonnegative Euclidean inputs=[27, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 35, 0, 27], [35, 27, 1, 8], [27, 8, 3, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=27 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative28:original nonnegative Euclidean inputs=[28, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 35, 0, 28], [35, 28, 1, 7], [28, 7, 4, 0]];terminal positive gcd=7

right modulus:original representative k=28 in1..35;verified gcd(k,n)=7

coprime indicator=0;include representative=No

right modulus representative29:original nonnegative Euclidean inputs=[29, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 35, 0, 29], [35, 29, 1, 6], [29, 6, 4, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1

right modulus:original representative k=29 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative30:original nonnegative Euclidean inputs=[30, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 35, 0, 30], [35, 30, 1, 5], [30, 5, 6, 0]];terminal positive gcd=5

right modulus:original representative k=30 in1..35;verified gcd(k,n)=5

coprime indicator=0;include representative=No

right modulus representative31:original nonnegative Euclidean inputs=[31, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[31, 35, 0, 31], [35, 31, 1, 4], [31, 4, 7, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=31 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative32:original nonnegative Euclidean inputs=[32, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 35, 0, 32], [35, 32, 1, 3], [32, 3, 10, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=32 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative33:original nonnegative Euclidean inputs=[33, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[33, 35, 0, 33], [35, 33, 1, 2], [33, 2, 16, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=33 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative34:original nonnegative Euclidean inputs=[34, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 35, 0, 34], [35, 34, 1, 1], [34, 1, 34, 0]];terminal positive gcd=1

right modulus:original representative k=34 in1..35;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative35:original nonnegative Euclidean inputs=[35, 35]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 35, 1, 0]];terminal positive gcd=35

right modulus:original representative k=35 in1..35;verified gcd(k,n)=35

coprime indicator=0;include representative=No

right modulus exact coprime indicator count := sum(1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0)

right modulus exact coprime indicator count = 24

right modulus:complete representative range1..35;all selected coprime representatives=[1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34]

phi(35)=24;representatives_1_through_n

product modulus representative1:original nonnegative Euclidean inputs=[1, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 70, 0, 1], [70, 1, 70, 0]];terminal positive gcd=1

product modulus:original representative k=1 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative2:original nonnegative Euclidean inputs=[2, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 70, 0, 2], [70, 2, 35, 0]];terminal positive gcd=2

product modulus:original representative k=2 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative3:original nonnegative Euclidean inputs=[3, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 70, 0, 3], [70, 3, 23, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=3 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative4:original nonnegative Euclidean inputs=[4, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 70, 0, 4], [70, 4, 17, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=4 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative5:original nonnegative Euclidean inputs=[5, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 70, 0, 5], [70, 5, 14, 0]];terminal positive gcd=5

product modulus:original representative k=5 in1..70;verified gcd(k,n)=5

coprime indicator=0;include representative=No

product modulus representative6:original nonnegative Euclidean inputs=[6, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 70, 0, 6], [70, 6, 11, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=6 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative7:original nonnegative Euclidean inputs=[7, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 70, 0, 7], [70, 7, 10, 0]];terminal positive gcd=7

product modulus:original representative k=7 in1..70;verified gcd(k,n)=7

coprime indicator=0;include representative=No

product modulus representative8:original nonnegative Euclidean inputs=[8, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 70, 0, 8], [70, 8, 8, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=8 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative9:original nonnegative Euclidean inputs=[9, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 70, 0, 9], [70, 9, 7, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=9 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative10:original nonnegative Euclidean inputs=[10, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 70, 0, 10], [70, 10, 7, 0]];terminal positive gcd=10

product modulus:original representative k=10 in1..70;verified gcd(k,n)=10

coprime indicator=0;include representative=No

product modulus representative11:original nonnegative Euclidean inputs=[11, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 70, 0, 11], [70, 11, 6, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=11 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative12:original nonnegative Euclidean inputs=[12, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 70, 0, 12], [70, 12, 5, 10], [12, 10, 1, 2], [10, 2, 5, 0]];terminal positive gcd=2

product modulus:original representative k=12 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative13:original nonnegative Euclidean inputs=[13, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 70, 0, 13], [70, 13, 5, 5], [13, 5, 2, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=13 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative14:original nonnegative Euclidean inputs=[14, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 70, 0, 14], [70, 14, 5, 0]];terminal positive gcd=14

product modulus:original representative k=14 in1..70;verified gcd(k,n)=14

coprime indicator=0;include representative=No

product modulus representative15:original nonnegative Euclidean inputs=[15, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 70, 0, 15], [70, 15, 4, 10], [15, 10, 1, 5], [10, 5, 2, 0]];terminal positive gcd=5

product modulus:original representative k=15 in1..70;verified gcd(k,n)=5

coprime indicator=0;include representative=No

product modulus representative16:original nonnegative Euclidean inputs=[16, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 70, 0, 16], [70, 16, 4, 6], [16, 6, 2, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=16 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative17:original nonnegative Euclidean inputs=[17, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 70, 0, 17], [70, 17, 4, 2], [17, 2, 8, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=17 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative18:original nonnegative Euclidean inputs=[18, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 70, 0, 18], [70, 18, 3, 16], [18, 16, 1, 2], [16, 2, 8, 0]];terminal positive gcd=2

product modulus:original representative k=18 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative19:original nonnegative Euclidean inputs=[19, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 70, 0, 19], [70, 19, 3, 13], [19, 13, 1, 6], [13, 6, 2, 1], [6, 1, 6, 0]];terminal positive gcd=1

product modulus:original representative k=19 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative20:original nonnegative Euclidean inputs=[20, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 70, 0, 20], [70, 20, 3, 10], [20, 10, 2, 0]];terminal positive gcd=10

product modulus:original representative k=20 in1..70;verified gcd(k,n)=10

coprime indicator=0;include representative=No

product modulus representative21:original nonnegative Euclidean inputs=[21, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 70, 0, 21], [70, 21, 3, 7], [21, 7, 3, 0]];terminal positive gcd=7

product modulus:original representative k=21 in1..70;verified gcd(k,n)=7

coprime indicator=0;include representative=No

product modulus representative22:original nonnegative Euclidean inputs=[22, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 70, 0, 22], [70, 22, 3, 4], [22, 4, 5, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=22 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative23:original nonnegative Euclidean inputs=[23, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 70, 0, 23], [70, 23, 3, 1], [23, 1, 23, 0]];terminal positive gcd=1

product modulus:original representative k=23 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative24:original nonnegative Euclidean inputs=[24, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 70, 0, 24], [70, 24, 2, 22], [24, 22, 1, 2], [22, 2, 11, 0]];terminal positive gcd=2

product modulus:original representative k=24 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative25:original nonnegative Euclidean inputs=[25, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 70, 0, 25], [70, 25, 2, 20], [25, 20, 1, 5], [20, 5, 4, 0]];terminal positive gcd=5

product modulus:original representative k=25 in1..70;verified gcd(k,n)=5

coprime indicator=0;include representative=No

product modulus representative26:original nonnegative Euclidean inputs=[26, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 70, 0, 26], [70, 26, 2, 18], [26, 18, 1, 8], [18, 8, 2, 2], [8, 2, 4, 0]];terminal positive gcd=2

product modulus:original representative k=26 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative27:original nonnegative Euclidean inputs=[27, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 70, 0, 27], [70, 27, 2, 16], [27, 16, 1, 11], [16, 11, 1, 5], [11, 5, 2, 1], [5, 1, 5, 0]];terminal positive gcd=1

product modulus:original representative k=27 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative28:original nonnegative Euclidean inputs=[28, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 70, 0, 28], [70, 28, 2, 14], [28, 14, 2, 0]];terminal positive gcd=14

product modulus:original representative k=28 in1..70;verified gcd(k,n)=14

coprime indicator=0;include representative=No

product modulus representative29:original nonnegative Euclidean inputs=[29, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 70, 0, 29], [70, 29, 2, 12], [29, 12, 2, 5], [12, 5, 2, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=29 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative30:original nonnegative Euclidean inputs=[30, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 70, 0, 30], [70, 30, 2, 10], [30, 10, 3, 0]];terminal positive gcd=10

product modulus:original representative k=30 in1..70;verified gcd(k,n)=10

coprime indicator=0;include representative=No

product modulus representative31:original nonnegative Euclidean inputs=[31, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[31, 70, 0, 31], [70, 31, 2, 8], [31, 8, 3, 7], [8, 7, 1, 1], [7, 1, 7, 0]];terminal positive gcd=1

product modulus:original representative k=31 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative32:original nonnegative Euclidean inputs=[32, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 70, 0, 32], [70, 32, 2, 6], [32, 6, 5, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=32 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative33:original nonnegative Euclidean inputs=[33, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[33, 70, 0, 33], [70, 33, 2, 4], [33, 4, 8, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=33 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative34:original nonnegative Euclidean inputs=[34, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 70, 0, 34], [70, 34, 2, 2], [34, 2, 17, 0]];terminal positive gcd=2

product modulus:original representative k=34 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative35:original nonnegative Euclidean inputs=[35, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 70, 0, 35], [70, 35, 2, 0]];terminal positive gcd=35

product modulus:original representative k=35 in1..70;verified gcd(k,n)=35

coprime indicator=0;include representative=No

product modulus representative36:original nonnegative Euclidean inputs=[36, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[36, 70, 0, 36], [70, 36, 1, 34], [36, 34, 1, 2], [34, 2, 17, 0]];terminal positive gcd=2

product modulus:original representative k=36 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative37:original nonnegative Euclidean inputs=[37, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[37, 70, 0, 37], [70, 37, 1, 33], [37, 33, 1, 4], [33, 4, 8, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=37 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative38:original nonnegative Euclidean inputs=[38, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[38, 70, 0, 38], [70, 38, 1, 32], [38, 32, 1, 6], [32, 6, 5, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=38 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative39:original nonnegative Euclidean inputs=[39, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[39, 70, 0, 39], [70, 39, 1, 31], [39, 31, 1, 8], [31, 8, 3, 7], [8, 7, 1, 1], [7, 1, 7, 0]];terminal positive gcd=1

product modulus:original representative k=39 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative40:original nonnegative Euclidean inputs=[40, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[40, 70, 0, 40], [70, 40, 1, 30], [40, 30, 1, 10], [30, 10, 3, 0]];terminal positive gcd=10

product modulus:original representative k=40 in1..70;verified gcd(k,n)=10

coprime indicator=0;include representative=No

product modulus representative41:original nonnegative Euclidean inputs=[41, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[41, 70, 0, 41], [70, 41, 1, 29], [41, 29, 1, 12], [29, 12, 2, 5], [12, 5, 2, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=41 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative42:original nonnegative Euclidean inputs=[42, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[42, 70, 0, 42], [70, 42, 1, 28], [42, 28, 1, 14], [28, 14, 2, 0]];terminal positive gcd=14

product modulus:original representative k=42 in1..70;verified gcd(k,n)=14

coprime indicator=0;include representative=No

product modulus representative43:original nonnegative Euclidean inputs=[43, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[43, 70, 0, 43], [70, 43, 1, 27], [43, 27, 1, 16], [27, 16, 1, 11], [16, 11, 1, 5], [11, 5, 2, 1], [5, 1, 5, 0]];terminal positive gcd=1

product modulus:original representative k=43 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative44:original nonnegative Euclidean inputs=[44, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[44, 70, 0, 44], [70, 44, 1, 26], [44, 26, 1, 18], [26, 18, 1, 8], [18, 8, 2, 2], [8, 2, 4, 0]];terminal positive gcd=2

product modulus:original representative k=44 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative45:original nonnegative Euclidean inputs=[45, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[45, 70, 0, 45], [70, 45, 1, 25], [45, 25, 1, 20], [25, 20, 1, 5], [20, 5, 4, 0]];terminal positive gcd=5

product modulus:original representative k=45 in1..70;verified gcd(k,n)=5

coprime indicator=0;include representative=No

product modulus representative46:original nonnegative Euclidean inputs=[46, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[46, 70, 0, 46], [70, 46, 1, 24], [46, 24, 1, 22], [24, 22, 1, 2], [22, 2, 11, 0]];terminal positive gcd=2

product modulus:original representative k=46 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative47:original nonnegative Euclidean inputs=[47, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[47, 70, 0, 47], [70, 47, 1, 23], [47, 23, 2, 1], [23, 1, 23, 0]];terminal positive gcd=1

product modulus:original representative k=47 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative48:original nonnegative Euclidean inputs=[48, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[48, 70, 0, 48], [70, 48, 1, 22], [48, 22, 2, 4], [22, 4, 5, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=48 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative49:original nonnegative Euclidean inputs=[49, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[49, 70, 0, 49], [70, 49, 1, 21], [49, 21, 2, 7], [21, 7, 3, 0]];terminal positive gcd=7

product modulus:original representative k=49 in1..70;verified gcd(k,n)=7

coprime indicator=0;include representative=No

product modulus representative50:original nonnegative Euclidean inputs=[50, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[50, 70, 0, 50], [70, 50, 1, 20], [50, 20, 2, 10], [20, 10, 2, 0]];terminal positive gcd=10

product modulus:original representative k=50 in1..70;verified gcd(k,n)=10

coprime indicator=0;include representative=No

product modulus representative51:original nonnegative Euclidean inputs=[51, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[51, 70, 0, 51], [70, 51, 1, 19], [51, 19, 2, 13], [19, 13, 1, 6], [13, 6, 2, 1], [6, 1, 6, 0]];terminal positive gcd=1

product modulus:original representative k=51 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative52:original nonnegative Euclidean inputs=[52, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[52, 70, 0, 52], [70, 52, 1, 18], [52, 18, 2, 16], [18, 16, 1, 2], [16, 2, 8, 0]];terminal positive gcd=2

product modulus:original representative k=52 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative53:original nonnegative Euclidean inputs=[53, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[53, 70, 0, 53], [70, 53, 1, 17], [53, 17, 3, 2], [17, 2, 8, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=53 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative54:original nonnegative Euclidean inputs=[54, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[54, 70, 0, 54], [70, 54, 1, 16], [54, 16, 3, 6], [16, 6, 2, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=54 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative55:original nonnegative Euclidean inputs=[55, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[55, 70, 0, 55], [70, 55, 1, 15], [55, 15, 3, 10], [15, 10, 1, 5], [10, 5, 2, 0]];terminal positive gcd=5

product modulus:original representative k=55 in1..70;verified gcd(k,n)=5

coprime indicator=0;include representative=No

product modulus representative56:original nonnegative Euclidean inputs=[56, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[56, 70, 0, 56], [70, 56, 1, 14], [56, 14, 4, 0]];terminal positive gcd=14

product modulus:original representative k=56 in1..70;verified gcd(k,n)=14

coprime indicator=0;include representative=No

product modulus representative57:original nonnegative Euclidean inputs=[57, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[57, 70, 0, 57], [70, 57, 1, 13], [57, 13, 4, 5], [13, 5, 2, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=57 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative58:original nonnegative Euclidean inputs=[58, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[58, 70, 0, 58], [70, 58, 1, 12], [58, 12, 4, 10], [12, 10, 1, 2], [10, 2, 5, 0]];terminal positive gcd=2

product modulus:original representative k=58 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative59:original nonnegative Euclidean inputs=[59, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[59, 70, 0, 59], [70, 59, 1, 11], [59, 11, 5, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=59 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative60:original nonnegative Euclidean inputs=[60, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[60, 70, 0, 60], [70, 60, 1, 10], [60, 10, 6, 0]];terminal positive gcd=10

product modulus:original representative k=60 in1..70;verified gcd(k,n)=10

coprime indicator=0;include representative=No

product modulus representative61:original nonnegative Euclidean inputs=[61, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[61, 70, 0, 61], [70, 61, 1, 9], [61, 9, 6, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=61 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative62:original nonnegative Euclidean inputs=[62, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[62, 70, 0, 62], [70, 62, 1, 8], [62, 8, 7, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=62 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative63:original nonnegative Euclidean inputs=[63, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[63, 70, 0, 63], [70, 63, 1, 7], [63, 7, 9, 0]];terminal positive gcd=7

product modulus:original representative k=63 in1..70;verified gcd(k,n)=7

coprime indicator=0;include representative=No

product modulus representative64:original nonnegative Euclidean inputs=[64, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[64, 70, 0, 64], [70, 64, 1, 6], [64, 6, 10, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=64 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative65:original nonnegative Euclidean inputs=[65, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[65, 70, 0, 65], [70, 65, 1, 5], [65, 5, 13, 0]];terminal positive gcd=5

product modulus:original representative k=65 in1..70;verified gcd(k,n)=5

coprime indicator=0;include representative=No

product modulus representative66:original nonnegative Euclidean inputs=[66, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[66, 70, 0, 66], [70, 66, 1, 4], [66, 4, 16, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=66 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative67:original nonnegative Euclidean inputs=[67, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[67, 70, 0, 67], [70, 67, 1, 3], [67, 3, 22, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=67 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative68:original nonnegative Euclidean inputs=[68, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[68, 70, 0, 68], [70, 68, 1, 2], [68, 2, 34, 0]];terminal positive gcd=2

product modulus:original representative k=68 in1..70;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative69:original nonnegative Euclidean inputs=[69, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[69, 70, 0, 69], [70, 69, 1, 1], [69, 1, 69, 0]];terminal positive gcd=1

product modulus:original representative k=69 in1..70;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative70:original nonnegative Euclidean inputs=[70, 70]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[70, 70, 1, 0]];terminal positive gcd=70

product modulus:original representative k=70 in1..70;verified gcd(k,n)=70

coprime indicator=0;include representative=No

product modulus exact coprime indicator count := sum(1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0)

product modulus exact coprime indicator count = 24

product modulus:complete representative range1..70;all selected coprime representatives=[1, 3, 9, 11, 13, 17, 19, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 67, 69]

phi(70)=24;representatives_1_through_n

totient supplied instance product of counts := mul(1, 24)

totient supplied instance product of counts = 24

original coprime inputs=[2, 35];gcd=1;product modulus=70;exact totients=[1, 24, 24]

phi(m)*phi(n)=24;same supplied-instance equality=Yes;instance_only_with_coprimality

Reverse checks

• S224 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S223 · premise_and_result_audit — Recompute totient supplied instance product of counts with exact rational arithmetic, not rounded decimals.
• S222 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S221 · premise_and_result_audit — Recompute product modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
• S220 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S219 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S218 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S217 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S216 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S215 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S214 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S213 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S212 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S210 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S209 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S208 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S206 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S205 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S204 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S203 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S202 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S201 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S200 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S199 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S198 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S196 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S194 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S193 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S189 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S188 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S187 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S185 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S182 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S181 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S180 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S179 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S177 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S175 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S173 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S079 · premise_and_result_audit — Recompute right modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute left modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute totient multiplicativity original modulus product with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(2, 35, 1, 24, 24, 24, Yes, instance_only_with_coprimality)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.