Why when you subtract the sorted-digit version of x from x the result is always a multiple of 9?

Question

$d(x)$ is $x$ - (sorted-digit version of $x$)

$d(75) = 75-57 = 18 = 9*2$

$d(574) = 574-457 = 117 = 9*13$

$d(4321) = 4321-1234 = 3087 = 9*343$

Why does it happen? Is there a proof that this is true for all positive integers?

also, i could not find any number $x$ such that $d(x) =$ $9*109$, $9*208$, $9*219$, $9*307$, $9*318$, $9*329$, $9*406$, etc. Why?

Here is the first number $x$ found where $d(x)$ is the $n$th multiple of 9:

d(0) = 0 = 9x0
d(10) = 9 = 9x1
d(20) = 18 = 9x2
d(30) = 27 = 9x3
d(40) = 36 = 9x4
d(50) = 45 = 9x5
d(60) = 54 = 9x6
d(70) = 63 = 9x7
d(80) = 72 = 9x8
d(90) = 81 = 9x9
d(101) = 90 = 9x10
d(100) = 99 = 9x11
d(120) = 108 = 9x12
d(130) = 117 = 9x13
d(140) = 126 = 9x14
d(150) = 135 = 9x15
d(160) = 144 = 9x16
d(170) = 153 = 9x17
d(180) = 162 = 9x18
d(190) = 171 = 9x19
d(202) = 180 = 9x20
d(201) = 189 = 9x21
d(200) = 198 = 9x22
d(230) = 207 = 9x23
d(240) = 216 = 9x24
d(250) = 225 = 9x25
d(260) = 234 = 9x26
d(270) = 243 = 9x27
d(280) = 252 = 9x28
d(290) = 261 = 9x29
d(303) = 270 = 9x30
d(302) = 279 = 9x31
d(301) = 288 = 9x32
d(300) = 297 = 9x33
d(340) = 306 = 9x34
d(350) = 315 = 9x35
d(360) = 324 = 9x36
d(370) = 333 = 9x37
d(380) = 342 = 9x38
d(390) = 351 = 9x39
d(404) = 360 = 9x40
d(403) = 369 = 9x41
d(402) = 378 = 9x42
d(401) = 387 = 9x43
d(400) = 396 = 9x44
d(450) = 405 = 9x45
d(460) = 414 = 9x46
d(470) = 423 = 9x47
d(480) = 432 = 9x48
d(490) = 441 = 9x49
d(505) = 450 = 9x50
d(504) = 459 = 9x51
d(503) = 468 = 9x52
d(502) = 477 = 9x53
d(501) = 486 = 9x54
d(500) = 495 = 9x55
d(560) = 504 = 9x56
d(570) = 513 = 9x57
d(580) = 522 = 9x58
d(590) = 531 = 9x59
d(606) = 540 = 9x60
d(605) = 549 = 9x61
d(604) = 558 = 9x62
d(603) = 567 = 9x63
d(602) = 576 = 9x64
d(601) = 585 = 9x65
d(600) = 594 = 9x66
d(670) = 603 = 9x67
d(680) = 612 = 9x68
d(690) = 621 = 9x69
d(707) = 630 = 9x70
d(706) = 639 = 9x71
d(705) = 648 = 9x72
d(704) = 657 = 9x73
d(703) = 666 = 9x74
d(702) = 675 = 9x75
d(701) = 684 = 9x76
d(700) = 693 = 9x77
d(780) = 702 = 9x78
d(790) = 711 = 9x79
d(808) = 720 = 9x80
d(807) = 729 = 9x81
d(806) = 738 = 9x82
d(805) = 747 = 9x83
d(804) = 756 = 9x84
d(803) = 765 = 9x85
d(802) = 774 = 9x86
d(801) = 783 = 9x87
d(800) = 792 = 9x88
d(890) = 801 = 9x89
d(909) = 810 = 9x90
d(908) = 819 = 9x91
d(907) = 828 = 9x92
d(906) = 837 = 9x93
d(905) = 846 = 9x94
d(904) = 855 = 9x95
d(903) = 864 = 9x96
d(902) = 873 = 9x97
d(901) = 882 = 9x98
d(900) = 891 = 9x99
d(1011) = 900 = 9x100
d(1021) = 909 = 9x101
d(1031) = 918 = 9x102
d(1041) = 927 = 9x103
d(1051) = 936 = 9x104
d(1061) = 945 = 9x105
d(1071) = 954 = 9x106
d(1081) = 963 = 9x107
d(1091) = 972 = 9x108
d(???) = 981 = 9x109
d(1001) = 990 = 9x110
d(1000) = 999 = 9x111
d(1020) = 1008 = 9x112
d(1030) = 1017 = 9x113
d(1040) = 1026 = 9x114
d(1050) = 1035 = 9x115
d(1060) = 1044 = 9x116
d(1070) = 1053 = 9x117
d(1080) = 1062 = 9x118
d(1090) = 1071 = 9x119
d(1202) = 1080 = 9x120
d(1100) = 1089 = 9x121
d(1210) = 1098 = 9x122
d(1230) = 1107 = 9x123
d(1240) = 1116 = 9x124
d(1250) = 1125 = 9x125
d(1260) = 1134 = 9x126
d(1270) = 1143 = 9x127
d(1280) = 1152 = 9x128
d(1290) = 1161 = 9x129
d(1303) = 1170 = 9x130
d(1302) = 1179 = 9x131
d(1200) = 1188 = 9x132
d(1310) = 1197 = 9x133
d(1340) = 1206 = 9x134
d(1350) = 1215 = 9x135
d(1360) = 1224 = 9x136
d(1370) = 1233 = 9x137
d(1380) = 1242 = 9x138
d(1390) = 1251 = 9x139
d(1404) = 1260 = 9x140
d(1403) = 1269 = 9x141
d(1402) = 1278 = 9x142
d(1300) = 1287 = 9x143
d(1410) = 1296 = 9x144
d(1450) = 1305 = 9x145
d(1460) = 1314 = 9x146
d(1470) = 1323 = 9x147
d(1480) = 1332 = 9x148
d(1490) = 1341 = 9x149
d(1505) = 1350 = 9x150
d(1504) = 1359 = 9x151
d(1503) = 1368 = 9x152
d(1502) = 1377 = 9x153
d(1400) = 1386 = 9x154
d(1510) = 1395 = 9x155
d(1560) = 1404 = 9x156
d(1570) = 1413 = 9x157
d(1580) = 1422 = 9x158
d(1590) = 1431 = 9x159
d(1606) = 1440 = 9x160
d(1605) = 1449 = 9x161
d(1604) = 1458 = 9x162
d(1603) = 1467 = 9x163
d(1602) = 1476 = 9x164
d(1500) = 1485 = 9x165
d(1610) = 1494 = 9x166
d(1670) = 1503 = 9x167
d(1680) = 1512 = 9x168
d(1690) = 1521 = 9x169
d(1707) = 1530 = 9x170
d(1706) = 1539 = 9x171
d(1705) = 1548 = 9x172
d(1704) = 1557 = 9x173
d(1703) = 1566 = 9x174
d(1702) = 1575 = 9x175
d(1600) = 1584 = 9x176
d(1710) = 1593 = 9x177
d(1780) = 1602 = 9x178
d(1790) = 1611 = 9x179
d(1808) = 1620 = 9x180
d(1807) = 1629 = 9x181
d(1806) = 1638 = 9x182
d(1805) = 1647 = 9x183
d(1804) = 1656 = 9x184
d(1803) = 1665 = 9x185
d(1802) = 1674 = 9x186
d(1700) = 1683 = 9x187
d(1810) = 1692 = 9x188
d(1890) = 1701 = 9x189
d(1909) = 1710 = 9x190
d(1908) = 1719 = 9x191
d(1907) = 1728 = 9x192
d(1906) = 1737 = 9x193
d(1905) = 1746 = 9x194
d(1904) = 1755 = 9x195
d(1903) = 1764 = 9x196
d(1902) = 1773 = 9x197
d(1800) = 1782 = 9x198
d(1910) = 1791 = 9x199
d(2022) = 1800 = 9x200
d(2032) = 1809 = 9x201
d(2042) = 1818 = 9x202
d(2052) = 1827 = 9x203
d(2062) = 1836 = 9x204
d(2072) = 1845 = 9x205
d(2082) = 1854 = 9x206
d(2092) = 1863 = 9x207
d(???) = 1872 = 9x208
d(1900) = 1881 = 9x209
d(2012) = 1890 = 9x210
d(2011) = 1899 = 9x211
d(2031) = 1908 = 9x212
d(2041) = 1917 = 9x213
d(2051) = 1926 = 9x214
d(2061) = 1935 = 9x215
d(2071) = 1944 = 9x216
d(2081) = 1953 = 9x217
d(2091) = 1962 = 9x218
d(???) = 1971 = 9x219
d(2002) = 1980 = 9x220
d(2001) = 1989 = 9x221
d(2000) = 1998 = 9x222
d(2030) = 2007 = 9x223
d(2040) = 2016 = 9x224
d(2050) = 2025 = 9x225
d(2060) = 2034 = 9x226
d(2070) = 2043 = 9x227
d(2080) = 2052 = 9x228
d(2090) = 2061 = 9x229
d(2303) = 2070 = 9x230
d(2201) = 2079 = 9x231
d(2100) = 2088 = 9x232
d(2320) = 2097 = 9x233
d(2340) = 2106 = 9x234
d(2350) = 2115 = 9x235
d(2360) = 2124 = 9x236
d(2370) = 2133 = 9x237
d(2380) = 2142 = 9x238
d(2390) = 2151 = 9x239
d(2404) = 2160 = 9x240
d(2403) = 2169 = 9x241
d(2200) = 2178 = 9x242
d(2310) = 2187 = 9x243
d(2420) = 2196 = 9x244
d(2450) = 2205 = 9x245
d(2460) = 2214 = 9x246
d(2470) = 2223 = 9x247
d(2480) = 2232 = 9x248
d(2490) = 2241 = 9x249
d(2505) = 2250 = 9x250
d(2504) = 2259 = 9x251
d(2503) = 2268 = 9x252
d(2300) = 2277 = 9x253
d(2410) = 2286 = 9x254
d(2520) = 2295 = 9x255
d(2560) = 2304 = 9x256
d(2570) = 2313 = 9x257
d(2580) = 2322 = 9x258
d(2590) = 2331 = 9x259
d(2606) = 2340 = 9x260
d(2605) = 2349 = 9x261
d(2604) = 2358 = 9x262
d(2603) = 2367 = 9x263
d(2400) = 2376 = 9x264
d(2510) = 2385 = 9x265
d(2620) = 2394 = 9x266
d(2670) = 2403 = 9x267
d(2680) = 2412 = 9x268
d(2690) = 2421 = 9x269
d(2707) = 2430 = 9x270
d(2706) = 2439 = 9x271
d(2705) = 2448 = 9x272
d(2704) = 2457 = 9x273
d(2703) = 2466 = 9x274
d(2500) = 2475 = 9x275
d(2610) = 2484 = 9x276
d(2720) = 2493 = 9x277
d(2780) = 2502 = 9x278
d(2790) = 2511 = 9x279
d(2808) = 2520 = 9x280
d(2807) = 2529 = 9x281
d(2806) = 2538 = 9x282
d(2805) = 2547 = 9x283
d(2804) = 2556 = 9x284
d(2803) = 2565 = 9x285
d(2600) = 2574 = 9x286
d(2710) = 2583 = 9x287
d(2820) = 2592 = 9x288
d(2890) = 2601 = 9x289
d(2909) = 2610 = 9x290
d(2908) = 2619 = 9x291
d(2907) = 2628 = 9x292
d(2906) = 2637 = 9x293
d(2905) = 2646 = 9x294
d(2904) = 2655 = 9x295
d(2903) = 2664 = 9x296
d(2700) = 2673 = 9x297
d(2810) = 2682 = 9x298
d(2920) = 2691 = 9x299
d(3033) = 2700 = 9x300
d(3043) = 2709 = 9x301
d(3053) = 2718 = 9x302
d(3063) = 2727 = 9x303
d(3073) = 2736 = 9x304
d(3083) = 2745 = 9x305
d(3093) = 2754 = 9x306
d(???) = 2763 = 9x307
d(2800) = 2772 = 9x308
d(2910) = 2781 = 9x309
d(3023) = 2790 = 9x310
d(3022) = 2799 = 9x311
d(3042) = 2808 = 9x312
d(3052) = 2817 = 9x313
d(3062) = 2826 = 9x314
d(3072) = 2835 = 9x315
d(3082) = 2844 = 9x316
d(3092) = 2853 = 9x317
d(???) = 2862 = 9x318
d(2900) = 2871 = 9x319
d(3013) = 2880 = 9x320
d(3012) = 2889 = 9x321
d(3011) = 2898 = 9x322
d(3041) = 2907 = 9x323
d(3051) = 2916 = 9x324
d(3061) = 2925 = 9x325
d(3071) = 2934 = 9x326
d(3081) = 2943 = 9x327
d(3091) = 2952 = 9x328
d(???) = 2961 = 9x329
d(3003) = 2970 = 9x330
d(3002) = 2979 = 9x331
d(3001) = 2988 = 9x332
d(3000) = 2997 = 9x333
d(3040) = 3006 = 9x334
d(3050) = 3015 = 9x335
d(3060) = 3024 = 9x336
d(3070) = 3033 = 9x337
d(3080) = 3042 = 9x338
d(3090) = 3051 = 9x339
d(3404) = 3060 = 9x340
d(3302) = 3069 = 9x341
d(3201) = 3078 = 9x342
d(3100) = 3087 = 9x343
d(3430) = 3096 = 9x344
d(3450) = 3105 = 9x345
d(3460) = 3114 = 9x346
d(3470) = 3123 = 9x347
d(3480) = 3132 = 9x348
d(3490) = 3141 = 9x349
d(3505) = 3150 = 9x350
d(3504) = 3159 = 9x351
d(3301) = 3168 = 9x352
d(3200) = 3177 = 9x353
d(3420) = 3186 = 9x354
d(3530) = 3195 = 9x355
d(3560) = 3204 = 9x356
d(3570) = 3213 = 9x357
d(3580) = 3222 = 9x358
d(3590) = 3231 = 9x359
d(3606) = 3240 = 9x360
d(3605) = 3249 = 9x361
d(3604) = 3258 = 9x362
d(3300) = 3267 = 9x363
d(3410) = 3276 = 9x364
d(3520) = 3285 = 9x365
d(3630) = 3294 = 9x366
d(3670) = 3303 = 9x367
d(3680) = 3312 = 9x368
d(3690) = 3321 = 9x369
d(3707) = 3330 = 9x370
d(3706) = 3339 = 9x371
d(3705) = 3348 = 9x372
d(3704) = 3357 = 9x373
d(3400) = 3366 = 9x374
d(3510) = 3375 = 9x375
d(3620) = 3384 = 9x376
d(3730) = 3393 = 9x377
d(3780) = 3402 = 9x378
d(3790) = 3411 = 9x379
d(3808) = 3420 = 9x380
d(3807) = 3429 = 9x381
d(3806) = 3438 = 9x382
d(3805) = 3447 = 9x383
d(3804) = 3456 = 9x384
d(3500) = 3465 = 9x385
d(3610) = 3474 = 9x386
d(3720) = 3483 = 9x387
d(3830) = 3492 = 9x388
d(3890) = 3501 = 9x389
d(3909) = 3510 = 9x390
d(3908) = 3519 = 9x391
d(3907) = 3528 = 9x392
d(3906) = 3537 = 9x393
d(3905) = 3546 = 9x394
d(3904) = 3555 = 9x395
d(3600) = 3564 = 9x396
d(3710) = 3573 = 9x397
d(3820) = 3582 = 9x398
d(3930) = 3591 = 9x399
d(4044) = 3600 = 9x400
d(4054) = 3609 = 9x401
d(4064) = 3618 = 9x402
d(4074) = 3627 = 9x403
d(4084) = 3636 = 9x404
d(4094) = 3645 = 9x405
d(???) = 3654 = 9x406
d(3700) = 3663 = 9x407
d(3810) = 3672 = 9x408
d(3920) = 3681 = 9x409
d(4034) = 3690 = 9x410
d(4033) = 3699 = 9x411
d(4053) = 3708 = 9x412
d(4063) = 3717 = 9x413
d(4073) = 3726 = 9x414
d(4083) = 3735 = 9x415
d(4093) = 3744 = 9x416

related ... https://math.stackexchange.com/questions/425098/prove-casting-out-nines-of-an-integer-is-equivalent-to-that-integer-modulo-9 — GEdgar, May 04 '23 at 12:56
Nice bit of work. I notice that $d(1481)=d(481)$, so maybe there are other smaller numbers that work. I don't know why those other $d$'s are missing. — Empy2, May 04 '23 at 13:07
@Empy2 the code i used was wrong, fixed now, still there are some missing multiples of 9, such as 9 * 109, 208, 219, 307, 318, 329, 406... — Wagner Martins, May 04 '23 at 13:40
@BillDubuque what should OP do about the last question, which they address extensively and doesn't appear in your copy? — Empy2, May 06 '23 at 15:02
@Empy2 The links in the dupe explain casting out nines (elevens, etc) for general radix (base). — Bill Dubuque, May 06 '23 at 17:26
@BillDubuque this posts asks about multiples of nine that never equal $d(n)=n-$sorted$(n)$ — Empy2, May 07 '23 at 03:12
@Empy2 Sorted is a special case of permuted in the linked dupe. — Bill Dubuque, May 07 '23 at 03:23

Kandinskij · Answer 1 · 2023-05-04T13:15:30.423

0

Notice that exchanging the position of the digit $k$ with the digit $j$ in a number $a_n...a_k...a_j...a_0$ is like adding:

$$-10^ka_k-10^ja_j+10^ka_j+10^ja_k=a_k(10^j-10^k)+a_j(10^k-10^j)$$

Since a difference between powers of ten is a multiple of $9$, exhanging two ciphers simply adds a multiple of $9$. Since every permutation can be obtained by a finite number of exchanges, this completes the proof.

Moreover

$$a_k(10^j-10^k)+a_j(10^k-10^j)=10^j(a_k-a_k10^{k-j}+a_j10^{k-j}-a_j)=10^j(10^{k-j}-1)(a_j-a_k)=9\cdot 10^j (1+10+...+10^{k-j-1})(a_j-a_k).$$ So we are interested to the factor:

$$10^j (1+10+...+10^{k-j-1})(a_j-a_k)$$

We must have $j=0$ otherwise the unit cipher is $0$ (and we would want it to be $9$):

$$(1+10+...+10^{k-1})(a_0-a_k)$$

Clearly we are interested only on the unit digit so we can take modulo $10$:

$$a_k-a_0$$

The only possibility for we want $a_k-a_0=9$ or $a_k-a_0=-1$ so we get $a_k=9$ and $a_0=0$ or $a_k=a_0-1$.

So your conjecture is wrong, take for example $2910-2019$.

edited May 04 '23 at 13:15

answered May 04 '23 at 12:48

Kandinskij

3,709

but $2019$ is not the sorted version of $2910$. Am I missing something? – D S May 04 '23 at 13:25
d(2910) would be 2910 - 0192 = 2718 = 9x302 – Wagner Martins May 04 '23 at 13:42
wait, i don't understand, even if it d(2910) = 2910−2019, that would equal 891 which is 9*99 – Wagner Martins May 04 '23 at 13:45
Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque May 04 '23 at 16:26

Why when you subtract the sorted-digit version of x from x the result is always a multiple of 9?

1 Answers1