Question 16669: Totient Multiplicativity Requires the Coprimality Premise

Original question and givens

Positive integers m=2, n=39 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=39 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, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 39, 0, 2], [39, 2, 19, 1], [2, 1, 2, 0]];terminal positive gcd=1

totient multiplicativity original modulus product := mul(2, 39)

totient multiplicativity original modulus product = 78

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, 39]

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

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

coprime indicator=1;include representative=Yes

right modulus representative2:original nonnegative Euclidean inputs=[2, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 39, 0, 2], [39, 2, 19, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=2 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative3:original nonnegative Euclidean inputs=[3, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 39, 0, 3], [39, 3, 13, 0]];terminal positive gcd=3

right modulus:original representative k=3 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative4:original nonnegative Euclidean inputs=[4, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 39, 0, 4], [39, 4, 9, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=4 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative5:original nonnegative Euclidean inputs=[5, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 39, 0, 5], [39, 5, 7, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

right modulus:original representative k=5 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative6:original nonnegative Euclidean inputs=[6, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 39, 0, 6], [39, 6, 6, 3], [6, 3, 2, 0]];terminal positive gcd=3

right modulus:original representative k=6 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative7:original nonnegative Euclidean inputs=[7, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 39, 0, 7], [39, 7, 5, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=7 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative8:original nonnegative Euclidean inputs=[8, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 39, 0, 8], [39, 8, 4, 7], [8, 7, 1, 1], [7, 1, 7, 0]];terminal positive gcd=1

right modulus:original representative k=8 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative9:original nonnegative Euclidean inputs=[9, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 39, 0, 9], [39, 9, 4, 3], [9, 3, 3, 0]];terminal positive gcd=3

right modulus:original representative k=9 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative10:original nonnegative Euclidean inputs=[10, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 39, 0, 10], [39, 10, 3, 9], [10, 9, 1, 1], [9, 1, 9, 0]];terminal positive gcd=1

right modulus:original representative k=10 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative11:original nonnegative Euclidean inputs=[11, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 39, 0, 11], [39, 11, 3, 6], [11, 6, 1, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1

right modulus:original representative k=11 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative12:original nonnegative Euclidean inputs=[12, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 39, 0, 12], [39, 12, 3, 3], [12, 3, 4, 0]];terminal positive gcd=3

right modulus:original representative k=12 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative13:original nonnegative Euclidean inputs=[13, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 39, 0, 13], [39, 13, 3, 0]];terminal positive gcd=13

right modulus:original representative k=13 in1..39;verified gcd(k,n)=13

coprime indicator=0;include representative=No

right modulus representative14:original nonnegative Euclidean inputs=[14, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 39, 0, 14], [39, 14, 2, 11], [14, 11, 1, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=14 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative15:original nonnegative Euclidean inputs=[15, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 39, 0, 15], [39, 15, 2, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal positive gcd=3

right modulus:original representative k=15 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative16:original nonnegative Euclidean inputs=[16, 39]

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

right modulus:original representative k=16 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative17:original nonnegative Euclidean inputs=[17, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 39, 0, 17], [39, 17, 2, 5], [17, 5, 3, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=17 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative18:original nonnegative Euclidean inputs=[18, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 39, 0, 18], [39, 18, 2, 3], [18, 3, 6, 0]];terminal positive gcd=3

right modulus:original representative k=18 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative19:original nonnegative Euclidean inputs=[19, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 39, 0, 19], [39, 19, 2, 1], [19, 1, 19, 0]];terminal positive gcd=1

right modulus:original representative k=19 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative20:original nonnegative Euclidean inputs=[20, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 39, 0, 20], [39, 20, 1, 19], [20, 19, 1, 1], [19, 1, 19, 0]];terminal positive gcd=1

right modulus:original representative k=20 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative21:original nonnegative Euclidean inputs=[21, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 39, 0, 21], [39, 21, 1, 18], [21, 18, 1, 3], [18, 3, 6, 0]];terminal positive gcd=3

right modulus:original representative k=21 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative22:original nonnegative Euclidean inputs=[22, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 39, 0, 22], [39, 22, 1, 17], [22, 17, 1, 5], [17, 5, 3, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=22 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative23:original nonnegative Euclidean inputs=[23, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 39, 0, 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

right modulus:original representative k=23 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative24:original nonnegative Euclidean inputs=[24, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 39, 0, 24], [39, 24, 1, 15], [24, 15, 1, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal positive gcd=3

right modulus:original representative k=24 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative25:original nonnegative Euclidean inputs=[25, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 39, 0, 25], [39, 25, 1, 14], [25, 14, 1, 11], [14, 11, 1, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=25 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative26:original nonnegative Euclidean inputs=[26, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 39, 0, 26], [39, 26, 1, 13], [26, 13, 2, 0]];terminal positive gcd=13

right modulus:original representative k=26 in1..39;verified gcd(k,n)=13

coprime indicator=0;include representative=No

right modulus representative27:original nonnegative Euclidean inputs=[27, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 39, 0, 27], [39, 27, 1, 12], [27, 12, 2, 3], [12, 3, 4, 0]];terminal positive gcd=3

right modulus:original representative k=27 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative28:original nonnegative Euclidean inputs=[28, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 39, 0, 28], [39, 28, 1, 11], [28, 11, 2, 6], [11, 6, 1, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1

right modulus:original representative k=28 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative29:original nonnegative Euclidean inputs=[29, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 39, 0, 29], [39, 29, 1, 10], [29, 10, 2, 9], [10, 9, 1, 1], [9, 1, 9, 0]];terminal positive gcd=1

right modulus:original representative k=29 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative30:original nonnegative Euclidean inputs=[30, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 39, 0, 30], [39, 30, 1, 9], [30, 9, 3, 3], [9, 3, 3, 0]];terminal positive gcd=3

right modulus:original representative k=30 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative31:original nonnegative Euclidean inputs=[31, 39]

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

right modulus:original representative k=31 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative32:original nonnegative Euclidean inputs=[32, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 39, 0, 32], [39, 32, 1, 7], [32, 7, 4, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=32 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative33:original nonnegative Euclidean inputs=[33, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[33, 39, 0, 33], [39, 33, 1, 6], [33, 6, 5, 3], [6, 3, 2, 0]];terminal positive gcd=3

right modulus:original representative k=33 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative34:original nonnegative Euclidean inputs=[34, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 39, 0, 34], [39, 34, 1, 5], [34, 5, 6, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

right modulus:original representative k=34 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative35:original nonnegative Euclidean inputs=[35, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 39, 0, 35], [39, 35, 1, 4], [35, 4, 8, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=35 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative36:original nonnegative Euclidean inputs=[36, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[36, 39, 0, 36], [39, 36, 1, 3], [36, 3, 12, 0]];terminal positive gcd=3

right modulus:original representative k=36 in1..39;verified gcd(k,n)=3

coprime indicator=0;include representative=No

right modulus representative37:original nonnegative Euclidean inputs=[37, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[37, 39, 0, 37], [39, 37, 1, 2], [37, 2, 18, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=37 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

right modulus representative38:original nonnegative Euclidean inputs=[38, 39]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[38, 39, 0, 38], [39, 38, 1, 1], [38, 1, 38, 0]];terminal positive gcd=1

right modulus:original representative k=38 in1..39;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

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

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

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

coprime indicator=0;include representative=No

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

right modulus exact coprime indicator count = 24

right modulus:complete representative range1..39;all selected coprime representatives=[1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 17, 19, 20, 22, 23, 25, 28, 29, 31, 32, 34, 35, 37, 38]

phi(39)=24;representatives_1_through_n

product modulus representative1:original nonnegative Euclidean inputs=[1, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 78, 0, 1], [78, 1, 78, 0]];terminal positive gcd=1

product modulus:original representative k=1 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative2:original nonnegative Euclidean inputs=[2, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 78, 0, 2], [78, 2, 39, 0]];terminal positive gcd=2

product modulus:original representative k=2 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative3:original nonnegative Euclidean inputs=[3, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 78, 0, 3], [78, 3, 26, 0]];terminal positive gcd=3

product modulus:original representative k=3 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative4:original nonnegative Euclidean inputs=[4, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 78, 0, 4], [78, 4, 19, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=4 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative5:original nonnegative Euclidean inputs=[5, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 78, 0, 5], [78, 5, 15, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=5 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative6:original nonnegative Euclidean inputs=[6, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 78, 0, 6], [78, 6, 13, 0]];terminal positive gcd=6

product modulus:original representative k=6 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative7:original nonnegative Euclidean inputs=[7, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 78, 0, 7], [78, 7, 11, 1], [7, 1, 7, 0]];terminal positive gcd=1

product modulus:original representative k=7 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative8:original nonnegative Euclidean inputs=[8, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 78, 0, 8], [78, 8, 9, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=8 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative9:original nonnegative Euclidean inputs=[9, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 78, 0, 9], [78, 9, 8, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal positive gcd=3

product modulus:original representative k=9 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative10:original nonnegative Euclidean inputs=[10, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 78, 0, 10], [78, 10, 7, 8], [10, 8, 1, 2], [8, 2, 4, 0]];terminal positive gcd=2

product modulus:original representative k=10 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative11:original nonnegative Euclidean inputs=[11, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 78, 0, 11], [78, 11, 7, 1], [11, 1, 11, 0]];terminal positive gcd=1

product modulus:original representative k=11 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative12:original nonnegative Euclidean inputs=[12, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 78, 0, 12], [78, 12, 6, 6], [12, 6, 2, 0]];terminal positive gcd=6

product modulus:original representative k=12 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative13:original nonnegative Euclidean inputs=[13, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 78, 0, 13], [78, 13, 6, 0]];terminal positive gcd=13

product modulus:original representative k=13 in1..78;verified gcd(k,n)=13

coprime indicator=0;include representative=No

product modulus representative14:original nonnegative Euclidean inputs=[14, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 78, 0, 14], [78, 14, 5, 8], [14, 8, 1, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=14 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative15:original nonnegative Euclidean inputs=[15, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 78, 0, 15], [78, 15, 5, 3], [15, 3, 5, 0]];terminal positive gcd=3

product modulus:original representative k=15 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative16:original nonnegative Euclidean inputs=[16, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 78, 0, 16], [78, 16, 4, 14], [16, 14, 1, 2], [14, 2, 7, 0]];terminal positive gcd=2

product modulus:original representative k=16 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative17:original nonnegative Euclidean inputs=[17, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 78, 0, 17], [78, 17, 4, 10], [17, 10, 1, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=17 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative18:original nonnegative Euclidean inputs=[18, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 78, 0, 18], [78, 18, 4, 6], [18, 6, 3, 0]];terminal positive gcd=6

product modulus:original representative k=18 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative19:original nonnegative Euclidean inputs=[19, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 78, 0, 19], [78, 19, 4, 2], [19, 2, 9, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=19 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative20:original nonnegative Euclidean inputs=[20, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 78, 0, 20], [78, 20, 3, 18], [20, 18, 1, 2], [18, 2, 9, 0]];terminal positive gcd=2

product modulus:original representative k=20 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative21:original nonnegative Euclidean inputs=[21, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 78, 0, 21], [78, 21, 3, 15], [21, 15, 1, 6], [15, 6, 2, 3], [6, 3, 2, 0]];terminal positive gcd=3

product modulus:original representative k=21 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative22:original nonnegative Euclidean inputs=[22, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 78, 0, 22], [78, 22, 3, 12], [22, 12, 1, 10], [12, 10, 1, 2], [10, 2, 5, 0]];terminal positive gcd=2

product modulus:original representative k=22 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative23:original nonnegative Euclidean inputs=[23, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 78, 0, 23], [78, 23, 3, 9], [23, 9, 2, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=23 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative24:original nonnegative Euclidean inputs=[24, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 78, 0, 24], [78, 24, 3, 6], [24, 6, 4, 0]];terminal positive gcd=6

product modulus:original representative k=24 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative25:original nonnegative Euclidean inputs=[25, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 78, 0, 25], [78, 25, 3, 3], [25, 3, 8, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=25 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative26:original nonnegative Euclidean inputs=[26, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 78, 0, 26], [78, 26, 3, 0]];terminal positive gcd=26

product modulus:original representative k=26 in1..78;verified gcd(k,n)=26

coprime indicator=0;include representative=No

product modulus representative27:original nonnegative Euclidean inputs=[27, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 78, 0, 27], [78, 27, 2, 24], [27, 24, 1, 3], [24, 3, 8, 0]];terminal positive gcd=3

product modulus:original representative k=27 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative28:original nonnegative Euclidean inputs=[28, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 78, 0, 28], [78, 28, 2, 22], [28, 22, 1, 6], [22, 6, 3, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=28 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative29:original nonnegative Euclidean inputs=[29, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 78, 0, 29], [78, 29, 2, 20], [29, 20, 1, 9], [20, 9, 2, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=29 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative30:original nonnegative Euclidean inputs=[30, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 78, 0, 30], [78, 30, 2, 18], [30, 18, 1, 12], [18, 12, 1, 6], [12, 6, 2, 0]];terminal positive gcd=6

product modulus:original representative k=30 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative31:original nonnegative Euclidean inputs=[31, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[31, 78, 0, 31], [78, 31, 2, 16], [31, 16, 1, 15], [16, 15, 1, 1], [15, 1, 15, 0]];terminal positive gcd=1

product modulus:original representative k=31 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative32:original nonnegative Euclidean inputs=[32, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 78, 0, 32], [78, 32, 2, 14], [32, 14, 2, 4], [14, 4, 3, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=32 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative33:original nonnegative Euclidean inputs=[33, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[33, 78, 0, 33], [78, 33, 2, 12], [33, 12, 2, 9], [12, 9, 1, 3], [9, 3, 3, 0]];terminal positive gcd=3

product modulus:original representative k=33 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative34:original nonnegative Euclidean inputs=[34, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 78, 0, 34], [78, 34, 2, 10], [34, 10, 3, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=34 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative35:original nonnegative Euclidean inputs=[35, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 78, 0, 35], [78, 35, 2, 8], [35, 8, 4, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=35 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative36:original nonnegative Euclidean inputs=[36, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[36, 78, 0, 36], [78, 36, 2, 6], [36, 6, 6, 0]];terminal positive gcd=6

product modulus:original representative k=36 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative37:original nonnegative Euclidean inputs=[37, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[37, 78, 0, 37], [78, 37, 2, 4], [37, 4, 9, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=37 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative38:original nonnegative Euclidean inputs=[38, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[38, 78, 0, 38], [78, 38, 2, 2], [38, 2, 19, 0]];terminal positive gcd=2

product modulus:original representative k=38 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative39:original nonnegative Euclidean inputs=[39, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[39, 78, 0, 39], [78, 39, 2, 0]];terminal positive gcd=39

product modulus:original representative k=39 in1..78;verified gcd(k,n)=39

coprime indicator=0;include representative=No

product modulus representative40:original nonnegative Euclidean inputs=[40, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[40, 78, 0, 40], [78, 40, 1, 38], [40, 38, 1, 2], [38, 2, 19, 0]];terminal positive gcd=2

product modulus:original representative k=40 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative41:original nonnegative Euclidean inputs=[41, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[41, 78, 0, 41], [78, 41, 1, 37], [41, 37, 1, 4], [37, 4, 9, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=41 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative42:original nonnegative Euclidean inputs=[42, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[42, 78, 0, 42], [78, 42, 1, 36], [42, 36, 1, 6], [36, 6, 6, 0]];terminal positive gcd=6

product modulus:original representative k=42 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative43:original nonnegative Euclidean inputs=[43, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[43, 78, 0, 43], [78, 43, 1, 35], [43, 35, 1, 8], [35, 8, 4, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=43 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative44:original nonnegative Euclidean inputs=[44, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[44, 78, 0, 44], [78, 44, 1, 34], [44, 34, 1, 10], [34, 10, 3, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=44 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative45:original nonnegative Euclidean inputs=[45, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[45, 78, 0, 45], [78, 45, 1, 33], [45, 33, 1, 12], [33, 12, 2, 9], [12, 9, 1, 3], [9, 3, 3, 0]];terminal positive gcd=3

product modulus:original representative k=45 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative46:original nonnegative Euclidean inputs=[46, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[46, 78, 0, 46], [78, 46, 1, 32], [46, 32, 1, 14], [32, 14, 2, 4], [14, 4, 3, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=46 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative47:original nonnegative Euclidean inputs=[47, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[47, 78, 0, 47], [78, 47, 1, 31], [47, 31, 1, 16], [31, 16, 1, 15], [16, 15, 1, 1], [15, 1, 15, 0]];terminal positive gcd=1

product modulus:original representative k=47 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative48:original nonnegative Euclidean inputs=[48, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[48, 78, 0, 48], [78, 48, 1, 30], [48, 30, 1, 18], [30, 18, 1, 12], [18, 12, 1, 6], [12, 6, 2, 0]];terminal positive gcd=6

product modulus:original representative k=48 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative49:original nonnegative Euclidean inputs=[49, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[49, 78, 0, 49], [78, 49, 1, 29], [49, 29, 1, 20], [29, 20, 1, 9], [20, 9, 2, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=49 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative50:original nonnegative Euclidean inputs=[50, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[50, 78, 0, 50], [78, 50, 1, 28], [50, 28, 1, 22], [28, 22, 1, 6], [22, 6, 3, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=50 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative51:original nonnegative Euclidean inputs=[51, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[51, 78, 0, 51], [78, 51, 1, 27], [51, 27, 1, 24], [27, 24, 1, 3], [24, 3, 8, 0]];terminal positive gcd=3

product modulus:original representative k=51 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative52:original nonnegative Euclidean inputs=[52, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[52, 78, 0, 52], [78, 52, 1, 26], [52, 26, 2, 0]];terminal positive gcd=26

product modulus:original representative k=52 in1..78;verified gcd(k,n)=26

coprime indicator=0;include representative=No

product modulus representative53:original nonnegative Euclidean inputs=[53, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[53, 78, 0, 53], [78, 53, 1, 25], [53, 25, 2, 3], [25, 3, 8, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=53 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative54:original nonnegative Euclidean inputs=[54, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[54, 78, 0, 54], [78, 54, 1, 24], [54, 24, 2, 6], [24, 6, 4, 0]];terminal positive gcd=6

product modulus:original representative k=54 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative55:original nonnegative Euclidean inputs=[55, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[55, 78, 0, 55], [78, 55, 1, 23], [55, 23, 2, 9], [23, 9, 2, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=55 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative56:original nonnegative Euclidean inputs=[56, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[56, 78, 0, 56], [78, 56, 1, 22], [56, 22, 2, 12], [22, 12, 1, 10], [12, 10, 1, 2], [10, 2, 5, 0]];terminal positive gcd=2

product modulus:original representative k=56 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative57:original nonnegative Euclidean inputs=[57, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[57, 78, 0, 57], [78, 57, 1, 21], [57, 21, 2, 15], [21, 15, 1, 6], [15, 6, 2, 3], [6, 3, 2, 0]];terminal positive gcd=3

product modulus:original representative k=57 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative58:original nonnegative Euclidean inputs=[58, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[58, 78, 0, 58], [78, 58, 1, 20], [58, 20, 2, 18], [20, 18, 1, 2], [18, 2, 9, 0]];terminal positive gcd=2

product modulus:original representative k=58 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative59:original nonnegative Euclidean inputs=[59, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[59, 78, 0, 59], [78, 59, 1, 19], [59, 19, 3, 2], [19, 2, 9, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=59 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative60:original nonnegative Euclidean inputs=[60, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[60, 78, 0, 60], [78, 60, 1, 18], [60, 18, 3, 6], [18, 6, 3, 0]];terminal positive gcd=6

product modulus:original representative k=60 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative61:original nonnegative Euclidean inputs=[61, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[61, 78, 0, 61], [78, 61, 1, 17], [61, 17, 3, 10], [17, 10, 1, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal positive gcd=1

product modulus:original representative k=61 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative62:original nonnegative Euclidean inputs=[62, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[62, 78, 0, 62], [78, 62, 1, 16], [62, 16, 3, 14], [16, 14, 1, 2], [14, 2, 7, 0]];terminal positive gcd=2

product modulus:original representative k=62 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative63:original nonnegative Euclidean inputs=[63, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[63, 78, 0, 63], [78, 63, 1, 15], [63, 15, 4, 3], [15, 3, 5, 0]];terminal positive gcd=3

product modulus:original representative k=63 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative64:original nonnegative Euclidean inputs=[64, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[64, 78, 0, 64], [78, 64, 1, 14], [64, 14, 4, 8], [14, 8, 1, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=64 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative65:original nonnegative Euclidean inputs=[65, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[65, 78, 0, 65], [78, 65, 1, 13], [65, 13, 5, 0]];terminal positive gcd=13

product modulus:original representative k=65 in1..78;verified gcd(k,n)=13

coprime indicator=0;include representative=No

product modulus representative66:original nonnegative Euclidean inputs=[66, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[66, 78, 0, 66], [78, 66, 1, 12], [66, 12, 5, 6], [12, 6, 2, 0]];terminal positive gcd=6

product modulus:original representative k=66 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative67:original nonnegative Euclidean inputs=[67, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[67, 78, 0, 67], [78, 67, 1, 11], [67, 11, 6, 1], [11, 1, 11, 0]];terminal positive gcd=1

product modulus:original representative k=67 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative68:original nonnegative Euclidean inputs=[68, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[68, 78, 0, 68], [78, 68, 1, 10], [68, 10, 6, 8], [10, 8, 1, 2], [8, 2, 4, 0]];terminal positive gcd=2

product modulus:original representative k=68 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative69:original nonnegative Euclidean inputs=[69, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[69, 78, 0, 69], [78, 69, 1, 9], [69, 9, 7, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal positive gcd=3

product modulus:original representative k=69 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative70:original nonnegative Euclidean inputs=[70, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[70, 78, 0, 70], [78, 70, 1, 8], [70, 8, 8, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=70 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative71:original nonnegative Euclidean inputs=[71, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[71, 78, 0, 71], [78, 71, 1, 7], [71, 7, 10, 1], [7, 1, 7, 0]];terminal positive gcd=1

product modulus:original representative k=71 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative72:original nonnegative Euclidean inputs=[72, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[72, 78, 0, 72], [78, 72, 1, 6], [72, 6, 12, 0]];terminal positive gcd=6

product modulus:original representative k=72 in1..78;verified gcd(k,n)=6

coprime indicator=0;include representative=No

product modulus representative73:original nonnegative Euclidean inputs=[73, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[73, 78, 0, 73], [78, 73, 1, 5], [73, 5, 14, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=73 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative74:original nonnegative Euclidean inputs=[74, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[74, 78, 0, 74], [78, 74, 1, 4], [74, 4, 18, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=74 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative75:original nonnegative Euclidean inputs=[75, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[75, 78, 0, 75], [78, 75, 1, 3], [75, 3, 25, 0]];terminal positive gcd=3

product modulus:original representative k=75 in1..78;verified gcd(k,n)=3

coprime indicator=0;include representative=No

product modulus representative76:original nonnegative Euclidean inputs=[76, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[76, 78, 0, 76], [78, 76, 1, 2], [76, 2, 38, 0]];terminal positive gcd=2

product modulus:original representative k=76 in1..78;verified gcd(k,n)=2

coprime indicator=0;include representative=No

product modulus representative77:original nonnegative Euclidean inputs=[77, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[77, 78, 0, 77], [78, 77, 1, 1], [77, 1, 77, 0]];terminal positive gcd=1

product modulus:original representative k=77 in1..78;verified gcd(k,n)=1

coprime indicator=1;include representative=Yes

product modulus representative78:original nonnegative Euclidean inputs=[78, 78]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[78, 78, 1, 0]];terminal positive gcd=78

product modulus:original representative k=78 in1..78;verified gcd(k,n)=78

coprime indicator=0;include representative=No

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

product modulus exact coprime indicator count = 24

product modulus:complete representative range1..78;all selected coprime representatives=[1, 5, 7, 11, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 77]

phi(78)=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, 39];gcd=1;product modulus=78;exact totients=[1, 24, 24]

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

Reverse checks

• S248 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S247 · premise_and_result_audit — Recompute totient supplied instance product of counts with exact rational arithmetic, not rounded decimals.
• S246 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S245 · premise_and_result_audit — Recompute product modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S243 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S242 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S241 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S240 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S239 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S238 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S237 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S236 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S235 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S234 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S233 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S232 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S231 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S230 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S229 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S228 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S227 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S226 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S225 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute right modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
• 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 — 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, 39, 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.