Question 16665: Totient Multiplicativity Requires the Coprimality Premise

Original question and givens

Positive integers m=2, n=31 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=31 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, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 31, 0, 2], [31, 2, 15, 1], [2, 1, 2, 0]];terminal positive gcd=1

totient multiplicativity original modulus product := mul(2, 31)

totient multiplicativity original modulus product = 62

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, 31]

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

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

coprime indicator=1;include representative=Yes

right modulus representative2:original nonnegative Euclidean inputs=[2, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 31, 0, 2], [31, 2, 15, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=2 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative3:original nonnegative Euclidean inputs=[3, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 31, 0, 3], [31, 3, 10, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=3 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative4:original nonnegative Euclidean inputs=[4, 31]

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

right modulus:original representative k=4 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative5:original nonnegative Euclidean inputs=[5, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 31, 0, 5], [31, 5, 6, 1], [5, 1, 5, 0]];terminal positive gcd=1

right modulus:original representative k=5 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative6:original nonnegative Euclidean inputs=[6, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 31, 0, 6], [31, 6, 5, 1], [6, 1, 6, 0]];terminal positive gcd=1

right modulus:original representative k=6 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative7:original nonnegative Euclidean inputs=[7, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 31, 0, 7], [31, 7, 4, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=7 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative8:original nonnegative Euclidean inputs=[8, 31]

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

right modulus:original representative k=8 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative9:original nonnegative Euclidean inputs=[9, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 31, 0, 9], [31, 9, 3, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal positive gcd=1

right modulus:original representative k=9 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative10:original nonnegative Euclidean inputs=[10, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 31, 0, 10], [31, 10, 3, 1], [10, 1, 10, 0]];terminal positive gcd=1

right modulus:original representative k=10 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative11:original nonnegative Euclidean inputs=[11, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 31, 0, 11], [31, 11, 2, 9], [11, 9, 1, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=11 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative12:original nonnegative Euclidean inputs=[12, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 31, 0, 12], [31, 12, 2, 7], [12, 7, 1, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=12 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative13:original nonnegative Euclidean inputs=[13, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 31, 0, 13], [31, 13, 2, 5], [13, 5, 2, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=13 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative14:original nonnegative Euclidean inputs=[14, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 31, 0, 14], [31, 14, 2, 3], [14, 3, 4, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=14 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative15:original nonnegative Euclidean inputs=[15, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 31, 0, 15], [31, 15, 2, 1], [15, 1, 15, 0]];terminal positive gcd=1

right modulus:original representative k=15 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative16:original nonnegative Euclidean inputs=[16, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 31, 0, 16], [31, 16, 1, 15], [16, 15, 1, 1], [15, 1, 15, 0]];terminal positive gcd=1

right modulus:original representative k=16 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative17:original nonnegative Euclidean inputs=[17, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 31, 0, 17], [31, 17, 1, 14], [17, 14, 1, 3], [14, 3, 4, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=17 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative18:original nonnegative Euclidean inputs=[18, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 31, 0, 18], [31, 18, 1, 13], [18, 13, 1, 5], [13, 5, 2, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=18 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative19:original nonnegative Euclidean inputs=[19, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 31, 0, 19], [31, 19, 1, 12], [19, 12, 1, 7], [12, 7, 1, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=19 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative20:original nonnegative Euclidean inputs=[20, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 31, 0, 20], [31, 20, 1, 11], [20, 11, 1, 9], [11, 9, 1, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=20 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative21:original nonnegative Euclidean inputs=[21, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 31, 0, 21], [31, 21, 1, 10], [21, 10, 2, 1], [10, 1, 10, 0]];terminal positive gcd=1

right modulus:original representative k=21 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative22:original nonnegative Euclidean inputs=[22, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 31, 0, 22], [31, 22, 1, 9], [22, 9, 2, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal positive gcd=1

right modulus:original representative k=22 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative23:original nonnegative Euclidean inputs=[23, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 31, 0, 23], [31, 23, 1, 8], [23, 8, 2, 7], [8, 7, 1, 1], [7, 1, 7, 0]];terminal positive gcd=1

right modulus:original representative k=23 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative24:original nonnegative Euclidean inputs=[24, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 31, 0, 24], [31, 24, 1, 7], [24, 7, 3, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=24 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative25:original nonnegative Euclidean inputs=[25, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 31, 0, 25], [31, 25, 1, 6], [25, 6, 4, 1], [6, 1, 6, 0]];terminal positive gcd=1

right modulus:original representative k=25 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative26:original nonnegative Euclidean inputs=[26, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 31, 0, 26], [31, 26, 1, 5], [26, 5, 5, 1], [5, 1, 5, 0]];terminal positive gcd=1

right modulus:original representative k=26 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative27:original nonnegative Euclidean inputs=[27, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 31, 0, 27], [31, 27, 1, 4], [27, 4, 6, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=27 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative28:original nonnegative Euclidean inputs=[28, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 31, 0, 28], [31, 28, 1, 3], [28, 3, 9, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=28 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative29:original nonnegative Euclidean inputs=[29, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 31, 0, 29], [31, 29, 1, 2], [29, 2, 14, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=29 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative30:original nonnegative Euclidean inputs=[30, 31]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 31, 0, 30], [31, 30, 1, 1], [30, 1, 30, 0]];terminal positive gcd=1

right modulus:original representative k=30 in1..31;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

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

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

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

coprime indicator=0;include representative=No

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

right modulus exact coprime indicator count = 30

right modulus:complete representative range1..31;all selected coprime representatives=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]

phi(31)=30;representatives_1_through_n

product modulus representative1:original nonnegative Euclidean inputs=[1, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 62, 0, 1], [62, 1, 62, 0]];terminal positive gcd=1

product modulus:original representative k=1 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative2:original nonnegative Euclidean inputs=[2, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 62, 0, 2], [62, 2, 31, 0]];terminal positive gcd=2

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

coprime indicator=0;include representative=No

product modulus representative3:original nonnegative Euclidean inputs=[3, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 62, 0, 3], [62, 3, 20, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=3 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative4:original nonnegative Euclidean inputs=[4, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 62, 0, 4], [62, 4, 15, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=4 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative5:original nonnegative Euclidean inputs=[5, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 62, 0, 5], [62, 5, 12, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=5 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative6:original nonnegative Euclidean inputs=[6, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 62, 0, 6], [62, 6, 10, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=6 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative7:original nonnegative Euclidean inputs=[7, 62]

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

product modulus:original representative k=7 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative8:original nonnegative Euclidean inputs=[8, 62]

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

product modulus:original representative k=8 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative9:original nonnegative Euclidean inputs=[9, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 62, 0, 9], [62, 9, 6, 8], [9, 8, 1, 1], [8, 1, 8, 0]];terminal positive gcd=1

product modulus:original representative k=9 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative10:original nonnegative Euclidean inputs=[10, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 62, 0, 10], [62, 10, 6, 2], [10, 2, 5, 0]];terminal positive gcd=2

product modulus:original representative k=10 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative11:original nonnegative Euclidean inputs=[11, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 62, 0, 11], [62, 11, 5, 7], [11, 7, 1, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=11 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative12:original nonnegative Euclidean inputs=[12, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 62, 0, 12], [62, 12, 5, 2], [12, 2, 6, 0]];terminal positive gcd=2

product modulus:original representative k=12 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative13:original nonnegative Euclidean inputs=[13, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 62, 0, 13], [62, 13, 4, 10], [13, 10, 1, 3], [10, 3, 3, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=13 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative14:original nonnegative Euclidean inputs=[14, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 62, 0, 14], [62, 14, 4, 6], [14, 6, 2, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=14 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative15:original nonnegative Euclidean inputs=[15, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 62, 0, 15], [62, 15, 4, 2], [15, 2, 7, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=15 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative16:original nonnegative Euclidean inputs=[16, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 62, 0, 16], [62, 16, 3, 14], [16, 14, 1, 2], [14, 2, 7, 0]];terminal positive gcd=2

product modulus:original representative k=16 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative17:original nonnegative Euclidean inputs=[17, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 62, 0, 17], [62, 17, 3, 11], [17, 11, 1, 6], [11, 6, 1, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1

product modulus:original representative k=17 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative18:original nonnegative Euclidean inputs=[18, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 62, 0, 18], [62, 18, 3, 8], [18, 8, 2, 2], [8, 2, 4, 0]];terminal positive gcd=2

product modulus:original representative k=18 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative19:original nonnegative Euclidean inputs=[19, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 62, 0, 19], [62, 19, 3, 5], [19, 5, 3, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=19 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative20:original nonnegative Euclidean inputs=[20, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 62, 0, 20], [62, 20, 3, 2], [20, 2, 10, 0]];terminal positive gcd=2

product modulus:original representative k=20 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative21:original nonnegative Euclidean inputs=[21, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 62, 0, 21], [62, 21, 2, 20], [21, 20, 1, 1], [20, 1, 20, 0]];terminal positive gcd=1

product modulus:original representative k=21 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative22:original nonnegative Euclidean inputs=[22, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 62, 0, 22], [62, 22, 2, 18], [22, 18, 1, 4], [18, 4, 4, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=22 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative23:original nonnegative Euclidean inputs=[23, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 62, 0, 23], [62, 23, 2, 16], [23, 16, 1, 7], [16, 7, 2, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=23 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative24:original nonnegative Euclidean inputs=[24, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 62, 0, 24], [62, 24, 2, 14], [24, 14, 1, 10], [14, 10, 1, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=24 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative25:original nonnegative Euclidean inputs=[25, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 62, 0, 25], [62, 25, 2, 12], [25, 12, 2, 1], [12, 1, 12, 0]];terminal positive gcd=1

product modulus:original representative k=25 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative26:original nonnegative Euclidean inputs=[26, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 62, 0, 26], [62, 26, 2, 10], [26, 10, 2, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=26 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative27:original nonnegative Euclidean inputs=[27, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 62, 0, 27], [62, 27, 2, 8], [27, 8, 3, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=27 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative28:original nonnegative Euclidean inputs=[28, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 62, 0, 28], [62, 28, 2, 6], [28, 6, 4, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=28 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative29:original nonnegative Euclidean inputs=[29, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 62, 0, 29], [62, 29, 2, 4], [29, 4, 7, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=29 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative30:original nonnegative Euclidean inputs=[30, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 62, 0, 30], [62, 30, 2, 2], [30, 2, 15, 0]];terminal positive gcd=2

product modulus:original representative k=30 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative31:original nonnegative Euclidean inputs=[31, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[31, 62, 0, 31], [62, 31, 2, 0]];terminal positive gcd=31

product modulus:original representative k=31 in1..62;verified gcd(k,n)=31

coprime indicator=0;include representative=No

product modulus representative32:original nonnegative Euclidean inputs=[32, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 62, 0, 32], [62, 32, 1, 30], [32, 30, 1, 2], [30, 2, 15, 0]];terminal positive gcd=2

product modulus:original representative k=32 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative33:original nonnegative Euclidean inputs=[33, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[33, 62, 0, 33], [62, 33, 1, 29], [33, 29, 1, 4], [29, 4, 7, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=33 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative34:original nonnegative Euclidean inputs=[34, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 62, 0, 34], [62, 34, 1, 28], [34, 28, 1, 6], [28, 6, 4, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=34 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative35:original nonnegative Euclidean inputs=[35, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 62, 0, 35], [62, 35, 1, 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

product modulus:original representative k=35 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative36:original nonnegative Euclidean inputs=[36, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[36, 62, 0, 36], [62, 36, 1, 26], [36, 26, 1, 10], [26, 10, 2, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=36 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative37:original nonnegative Euclidean inputs=[37, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[37, 62, 0, 37], [62, 37, 1, 25], [37, 25, 1, 12], [25, 12, 2, 1], [12, 1, 12, 0]];terminal positive gcd=1

product modulus:original representative k=37 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative38:original nonnegative Euclidean inputs=[38, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[38, 62, 0, 38], [62, 38, 1, 24], [38, 24, 1, 14], [24, 14, 1, 10], [14, 10, 1, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=38 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative39:original nonnegative Euclidean inputs=[39, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[39, 62, 0, 39], [62, 39, 1, 23], [39, 23, 1, 16], [23, 16, 1, 7], [16, 7, 2, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=39 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative40:original nonnegative Euclidean inputs=[40, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[40, 62, 0, 40], [62, 40, 1, 22], [40, 22, 1, 18], [22, 18, 1, 4], [18, 4, 4, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=40 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative41:original nonnegative Euclidean inputs=[41, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[41, 62, 0, 41], [62, 41, 1, 21], [41, 21, 1, 20], [21, 20, 1, 1], [20, 1, 20, 0]];terminal positive gcd=1

product modulus:original representative k=41 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative42:original nonnegative Euclidean inputs=[42, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[42, 62, 0, 42], [62, 42, 1, 20], [42, 20, 2, 2], [20, 2, 10, 0]];terminal positive gcd=2

product modulus:original representative k=42 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative43:original nonnegative Euclidean inputs=[43, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[43, 62, 0, 43], [62, 43, 1, 19], [43, 19, 2, 5], [19, 5, 3, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=43 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative44:original nonnegative Euclidean inputs=[44, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[44, 62, 0, 44], [62, 44, 1, 18], [44, 18, 2, 8], [18, 8, 2, 2], [8, 2, 4, 0]];terminal positive gcd=2

product modulus:original representative k=44 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative45:original nonnegative Euclidean inputs=[45, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[45, 62, 0, 45], [62, 45, 1, 17], [45, 17, 2, 11], [17, 11, 1, 6], [11, 6, 1, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1

product modulus:original representative k=45 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative46:original nonnegative Euclidean inputs=[46, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[46, 62, 0, 46], [62, 46, 1, 16], [46, 16, 2, 14], [16, 14, 1, 2], [14, 2, 7, 0]];terminal positive gcd=2

product modulus:original representative k=46 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative47:original nonnegative Euclidean inputs=[47, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[47, 62, 0, 47], [62, 47, 1, 15], [47, 15, 3, 2], [15, 2, 7, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=47 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative48:original nonnegative Euclidean inputs=[48, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[48, 62, 0, 48], [62, 48, 1, 14], [48, 14, 3, 6], [14, 6, 2, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=48 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative49:original nonnegative Euclidean inputs=[49, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[49, 62, 0, 49], [62, 49, 1, 13], [49, 13, 3, 10], [13, 10, 1, 3], [10, 3, 3, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=49 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative50:original nonnegative Euclidean inputs=[50, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[50, 62, 0, 50], [62, 50, 1, 12], [50, 12, 4, 2], [12, 2, 6, 0]];terminal positive gcd=2

product modulus:original representative k=50 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative51:original nonnegative Euclidean inputs=[51, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[51, 62, 0, 51], [62, 51, 1, 11], [51, 11, 4, 7], [11, 7, 1, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=51 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative52:original nonnegative Euclidean inputs=[52, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[52, 62, 0, 52], [62, 52, 1, 10], [52, 10, 5, 2], [10, 2, 5, 0]];terminal positive gcd=2

product modulus:original representative k=52 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative53:original nonnegative Euclidean inputs=[53, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[53, 62, 0, 53], [62, 53, 1, 9], [53, 9, 5, 8], [9, 8, 1, 1], [8, 1, 8, 0]];terminal positive gcd=1

product modulus:original representative k=53 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative54:original nonnegative Euclidean inputs=[54, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[54, 62, 0, 54], [62, 54, 1, 8], [54, 8, 6, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=54 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative55:original nonnegative Euclidean inputs=[55, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[55, 62, 0, 55], [62, 55, 1, 7], [55, 7, 7, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal positive gcd=1

product modulus:original representative k=55 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative56:original nonnegative Euclidean inputs=[56, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[56, 62, 0, 56], [62, 56, 1, 6], [56, 6, 9, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=56 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative57:original nonnegative Euclidean inputs=[57, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[57, 62, 0, 57], [62, 57, 1, 5], [57, 5, 11, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=57 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative58:original nonnegative Euclidean inputs=[58, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[58, 62, 0, 58], [62, 58, 1, 4], [58, 4, 14, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=58 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative59:original nonnegative Euclidean inputs=[59, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[59, 62, 0, 59], [62, 59, 1, 3], [59, 3, 19, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=59 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative60:original nonnegative Euclidean inputs=[60, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[60, 62, 0, 60], [62, 60, 1, 2], [60, 2, 30, 0]];terminal positive gcd=2

product modulus:original representative k=60 in1..62;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative61:original nonnegative Euclidean inputs=[61, 62]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[61, 62, 0, 61], [62, 61, 1, 1], [61, 1, 61, 0]];terminal positive gcd=1

product modulus:original representative k=61 in1..62;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

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

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[62, 62, 1, 0]];terminal positive gcd=62

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

coprime indicator=0;include representative=No

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

product modulus exact coprime indicator count = 30

product modulus:complete representative range1..62;all selected coprime representatives=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61]

phi(62)=30;representatives_1_through_n

totient supplied instance product of counts := mul(1, 30)

totient supplied instance product of counts = 30

original coprime inputs=[2, 31];gcd=1;product modulus=62;exact totients=[1, 30, 30]

phi(m)*phi(n)=30;same supplied-instance equality=Yes;instance_only_with_coprimality

Reverse checks

• 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 — Recompute totient supplied instance product of counts with exact rational arithmetic, not rounded decimals.
• 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 — Recompute product modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute right modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
• 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, 31, 1, 30, 30, 30, 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.