I was struggling to find something that caught my fancy regarding the number associated with my diurnal age today: 27929. I thought I'd look at its reverse, 92972, and compare their factorisations. The results were:$$ \begin{align} 27929 = 11 \times 2539 \\ 92972 ==2^2 \times 11 \times 2113$$Clearly, the number and its reverse share a common prime factor of 11. I then realised that 27929 has a digit sum of 29 and the two digits, when added together, give 11. So I then decided to look for numbers with the following properties:
- number is divisible by 11
- its reverse is also divisible by 11
- its sum of digits gives a number whose digits sum to 11
It turns out that there are only 30 numbers that satisfy these criteria in the range up to 40000. They are:
20999, 21989, 22979, 23969, 24959, 25949, 26939, 27929, 28919, 29909, 30899, 30998, 31889, 31988, 32879, 32978, 33869, 33968, 34859, 34958, 35849, 35948, 36839, 36938, 37829, 37928, 38819, 38918, 39809, 39908
The details are (permalink):
number factors reverse factors digit sum sum
20999 11 * 23 * 83 99902 2 * 11 * 19 * 239 29 11
21989 11 * 1999 98912 2^5 * 11 * 281 29 11
22979 11 * 2089 97922 2 * 11 * 4451 29 11
23969 11 * 2179 96932 2^2 * 11 * 2203 29 11
24959 11 * 2269 95942 2 * 7^2 * 11 * 89 29 11
25949 7 * 11 * 337 94952 2^3 * 11 * 13 * 83 29 11
26939 11 * 31 * 79 93962 2 * 11 * 4271 29 11
27929 11 * 2539 92972 2^2 * 11 * 2113 29 11
28919 11^2 * 239 91982 2 * 11 * 37 * 113 29 11
29909 11 * 2719 90992 2^4 * 11^2 * 47 29 11
30899 11 * 53^2 99803 11 * 43 * 211 29 11
30998 2 * 11 * 1409 89903 11^2 * 743 29 11
31889 11 * 13 * 223 98813 11 * 13 * 691 29 11
31988 2^2 * 11 * 727 88913 11 * 59 * 137 29 11
32879 7^2 * 11 * 61 97823 11 * 8893 29 11
32978 2 * 11 * 1499 87923 11 * 7993 29 11
33869 11 * 3079 96833 11 * 8803 29 11
33968 2^4 * 11 * 193 86933 7 * 11 * 1129 29 11
34859 11 * 3169 95843 11 * 8713 29 11
34958 2 * 7 * 11 * 227 85943 11 * 13 * 601 29 11
35849 11 * 3259 94853 11 * 8623 29 11
35948 2^2 * 11 * 19 * 43 84953 11 * 7723 29 11
36839 11 * 17 * 197 93863 7 * 11 * 23 * 53 29 11
36938 2 * 11 * 23 * 73 83963 11 * 17 * 449 29 11
37829 11 * 19 * 181 92873 11 * 8443 29 11
37928 2^3 * 11 * 431 82973 11 * 19 * 397 29 11
38819 11 * 3529 91883 11 * 8353 29 11
38918 2 * 11 * 29 * 61 81983 11 * 29 * 257 29 11
39809 7 * 11^2 * 47 90893 11 * 8263 29 11
39908 2^2 * 11 * 907 80993 11 * 37 * 199 29 11
The algorithm can be modified to search for prime numbers other than 11. For example, there are 80 numbers in the range up to 40000 that satisfy these criteria:
- number is divisible by 7
- its reverse is also divisible by 7
- its sum of digits gives a number whose digits sum to 7
These numbers are (permalink):
259, 952, 1078, 1708, 2527, 2779, 3346, 3598, 4165, 5614, 5866, 6433, 6685, 7252, 8071, 8701, 8953, 9079, 9709, 9772, 10087, 10717, 10969, 11536, 11788, 12103, 12355, 13174, 13804, 14623, 14875, 15442, 15694, 17017, 17269, 17962, 18088, 18718, 19537, 19789, 20545, 20797, 21364, 22183, 22813, 23884, 24451, 25207, 25459, 26026, 26278, 26908, 26971, 27097, 27727, 27979, 28546, 28798, 29113, 29365, 30121, 30373, 31129, 31192, 31822, 32641, 32893, 33649, 34216, 34468, 35035, 35287, 35917, 36736, 36988, 37303, 37555, 38122, 38374, 39823
The details are (permalink):
number factors reverse factors digit sum sum
259 7 * 37 952 2^3 * 7 * 17 16 7
952 2^3 * 7 * 17 259 7 * 37 16 7
1078 2 * 7^2 * 11 8701 7 * 11 * 113 16 7
1708 2^2 * 7 * 61 8071 7 * 1153 16 7
2527 7 * 19^2 7252 2^2 * 7^2 * 37 16 7
2779 7 * 397 9772 2^2 * 7 * 349 25 7
3346 2 * 7 * 239 6433 7 * 919 16 7
3598 2 * 7 * 257 8953 7 * 1279 25 7
4165 5 * 7^2 * 17 5614 2 * 7 * 401 16 7
5614 2 * 7 * 401 4165 5 * 7^2 * 17 16 7
5866 2 * 7 * 419 6685 5 * 7 * 191 25 7
6433 7 * 919 3346 2 * 7 * 239 16 7
6685 5 * 7 * 191 5866 2 * 7 * 419 25 7
7252 2^2 * 7^2 * 37 2527 7 * 19^2 16 7
8071 7 * 1153 1708 2^2 * 7 * 61 16 7
8701 7 * 11 * 113 1078 2 * 7^2 * 11 16 7
8953 7 * 1279 3598 2 * 7 * 257 25 7
9079 7 * 1297 9709 7 * 19 * 73 25 7
9709 7 * 19 * 73 9079 7 * 1297 25 7
9772 2^2 * 7 * 349 2779 7 * 397 25 7
10087 7 * 11 * 131 78001 7 * 11 * 1013 16 7
10717 7 * 1531 71701 7 * 10243 16 7
10969 7 * 1567 96901 7 * 109 * 127 25 7
11536 2^4 * 7 * 103 63511 7 * 43 * 211 16 7
11788 2^2 * 7 * 421 88711 7 * 19 * 23 * 29 25 7
12103 7^2 * 13 * 19 30121 7 * 13 * 331 7 7
12355 5 * 7 * 353 55321 7^2 * 1129 16 7
13174 2 * 7 * 941 47131 7 * 6733 16 7
13804 2^2 * 7 * 17 * 29 40831 7 * 19 * 307 16 7
14623 7 * 2089 32641 7 * 4663 16 7
14875 5^3 * 7 * 17 57841 7 * 8263 25 7
15442 2 * 7 * 1103 24451 7^2 * 499 16 7
15694 2 * 7 * 19 * 59 49651 7 * 41 * 173 25 7
17017 7 * 11 * 13 * 17 71071 7 * 11 * 13 * 71 16 7
17269 7 * 2467 96271 7 * 17 * 809 25 7
17962 2 * 7 * 1283 26971 7 * 3853 25 7
18088 2^3 * 7 * 17 * 19 88081 7 * 12583 25 7
18718 2 * 7^2 * 191 81781 7^2 * 1669 25 7
19537 7 * 2791 73591 7 * 10513 25 7
19789 7 * 11 * 257 98791 7 * 11 * 1283 34 7
20545 5 * 7 * 587 54502 2 * 7 * 17 * 229 16 7
20797 7 * 2971 79702 2 * 7 * 5693 25 7
21364 2^2 * 7^2 * 109 46312 2^3 * 7 * 827 16 7
22183 7 * 3169 38122 2 * 7^2 * 389 16 7
22813 7 * 3259 31822 2 * 7 * 2273 16 7
23884 2^2 * 7 * 853 48832 2^6 * 7 * 109 25 7
24451 7^2 * 499 15442 2 * 7 * 1103 16 7
25207 7 * 13 * 277 70252 2^2 * 7 * 13 * 193 16 7
25459 7 * 3637 95452 2^2 * 7^2 * 487 25 7
26026 2 * 7 * 11 * 13^2 62062 2 * 7 * 11 * 13 * 31 16 7
26278 2 * 7 * 1877 87262 2 * 7 * 23 * 271 25 7
26908 2^2 * 7 * 31^2 80962 2 * 7 * 5783 25 7
26971 7 * 3853 17962 2 * 7 * 1283 25 7
27097 7^3 * 79 79072 2^5 * 7 * 353 25 7
27727 7 * 17 * 233 72772 2^2 * 7 * 23 * 113 25 7
27979 7^2 * 571 97972 2^2 * 7 * 3499 34 7
28546 2 * 7 * 2039 64582 2 * 7^2 * 659 25 7
28798 2 * 7 * 11^2 * 17 89782 2 * 7 * 11^2 * 53 34 7
29113 7 * 4159 31192 2^3 * 7 * 557 16 7
29365 5 * 7 * 839 56392 2^3 * 7 * 19 * 53 25 7
30121 7 * 13 * 331 12103 7^2 * 13 * 19 7 7
30373 7 * 4339 37303 7 * 73^2 16 7
31129 7 * 4447 92113 7 * 13159 16 7
31192 2^3 * 7 * 557 29113 7 * 4159 16 7
31822 2 * 7 * 2273 22813 7 * 3259 16 7
32641 7 * 4663 14623 7 * 2089 16 7
32893 7 * 37 * 127 39823 7 * 5689 25 7
33649 7 * 11 * 19 * 23 94633 7 * 11 * 1229 25 7
34216 2^3 * 7 * 13 * 47 61243 7 * 13 * 673 16 7
34468 2^2 * 7 * 1231 86443 7 * 53 * 233 25 7
35035 5 * 7^2 * 11 * 13 53053 7 * 11 * 13 * 53 16 7
35287 7 * 71^2 78253 7^2 * 1597 25 7
35917 7^2 * 733 71953 7 * 19 * 541 25 7
36736 2^7 * 7 * 41 63763 7 * 9109 25 7
36988 2^2 * 7 * 1321 88963 7 * 71 * 179 34 7
37303 7 * 73^2 30373 7 * 4339 16 7
37555 5 * 7 * 29 * 37 55573 7 * 17 * 467 25 7
38122 2 * 7^2 * 389 22183 7 * 3169 16 7
38374 2 * 7 * 2741 47383 7^2 * 967 25 7
39823 7 * 5689 32893 7 * 37 * 127 25 7
No comments:
Post a Comment