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Deux Duodecimal Prime Number Determination. (Posted on 2023-03-24) Difficulty: 3 of 5
N is a 5-digit duodecimal prime number which consists of precisely two distinct nonzero base-12 digits.

Determine the smallest and largest value of N.

*** N cannot admit of any leading zeros.

See The Solution Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 2
The smallest and largest are 10011 and bbbb7, corresponding to decimal 20749 and 248827 respectively.

There are 172 altogether:

    base
  12     10
10011  20749
11151  22669
11177  22699
11191  22717
111b1  22741
111bb  22751
11221  22777
11411  23053
11511  23197
11515  23201
11771  23557
11777  23563
11811  23629
11911  23773
11991  23869
11a11  23917
11b11  24061
11b1b  24071
11bb1  24181
15115  29537
15151  29581
15511  30109
15515  30113
15555  30161
16161  31321
16611  31981
17777  33931
18881  35809
19191  36541
19991  37693
1a1a1  38281
1b111  39901
1b1bb  40031
1bb11  41341
1bb1b  41351
22227  45247
22255  45281
22277  45307
22525  45677
22777  46027
27227  53887
27777  54667
2bb2b  62099
2bbbb  62207
31311  64381
33337  67867
333bb  67967
33737  68443
33777  68491
33bbb  69119
35555  71633
37337  74779
37377  74827
37777  75403
3bb3b  82847
41441  85297
44441  90481
44477  90523
44555  90641
44b4b  91499
44bbb  91583
45545  92357
45555  92369
47477  95707
4b44b 102587
51151 105613
51551 106189
52225 107453
55115 112481
55535 113081
55545 113093
5555b 113111
55565 113117
55577 113131
55595 113153
555b5 113177
55775 113417
55855 113537
55b55 113969
55bb5 114041
56555 114833
57755 116849
58885 118757
59555 120017
59995 120641
5b55b 123479
5b5bb 123551
5bb55 124337
5bb5b 124343
61661 127081
65565 133853
66565 135581
66611 135661
66661 135721
66667 135727
6666b 135731
66bbb 136511
6bb6b 145091
72227 148927
75557 154579
76777 156619
77117 157411
77171 157477
77177 157483
77277 157627
77377 157771
77711 158269
77771 158341
7777b 158351
77787 158359
77797 158371
77b77 158923
77b7b 158927
78787 160087
78887 160231
79977 162091
7a7a7 163567
7b7bb 165311
7bbb7 165883
7bbbb 165887
81811 168781
85585 175349
85885 175781
87777 179083
88181 179953
88555 180497
88585 180533
88777 180811
88bbb 181439
8b8bb 186191
8bb8b 186587
91911 189661
91991 189757
97777 199819
97997 200131
99977 203563
9999b 203591
a11a1 209353
aa111 224797
aaaa1 226201
aabbb 226367
ababb 227951
b1111 229981
b11b1 230101
b1b1b 231431
b1bbb 231551
b2bbb 233279
b333b 233759
b3bbb 235007
b5bbb 238463
b6b6b 240131
b7777 241291
b77bb 241343
b7b77 241867
b7bbb 241919
b8bbb 243647
b99bb 245087
baaab 246947
bb1b1 247381
bb1bb 247391
bb44b 247739
bb555 247889
bb66b 248051
bb777 248203
bb9bb 248543
bbb11 248701
bbb2b 248723
bbb55 248753
bbb77 248779
bbb7b 248783
bbbb1 248821
bbbb7 248827
 
 
found by

clearvars,clc
d='0123456789ab'; ct=0; solNo={};
for c1=1:12
  d1=d(c1);
  for c2=1:12
    d2=d(c2);
    if c1~=c2
      for ch=1:2
        idx=nchoosek(1:5,ch);
        for i=1:length(idx)
          nd=repmat(d2,1,5);
          nd(idx(i,:))=d1;
          p=base2dec(nd,12);
          if isprime(p) && nd(1)~='0'
            ct=ct+1;
            solNo{end+1}=[char(nd) ' ' char(string(p))];
          end
        end
      end
    end
  end
end
solNo=sort(solNo);
for i=1:length(solNo)
   disp(solNo{i})
end

  Posted by Charlie on 2023-03-24 08:27:06
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