Modest Numbers

Modest numbers are a category of numbers that I haven't paid a great deal of attention to. They are defined by Numbers Aplenty in the following manner:

A number \(n\) is called modest if its digits can be separated into two numbers \(a\) and \(b\) such that \(n\) divided by \(b\) gives \(a\) as remainder.

The reason its came to my attention is that Julian Day Number 2459999 is approaching and it has the property of being a modest number because 9999 divides 2459999 to give 245 or to put it in another format:$$2459999 \! \! \mod 99999 \equiv 245$$I developed an algorithm in SageMath to determine all the modest numbers in a given range. Here are the modest numbers in the range up to 40,000 (permalink), constituting 2.25% of the range:

13, 19, 23, 26, 29, 39, 46, 49, 59, 69, 79, 89, 103, 109, 111, 133, 199, 203, 206, 209, 211, 218, 222, 233, 266, 299, 309, 311, 327, 333, 399, 406, 409, 411, 412, 418, 422, 433, 436, 444, 466, 499, 509, 511, 515, 533, 545, 555, 599, 609, 611, 618, 622, 627, 633, 654, 666, 699, 709, 711, 721, 733, 763, 777, 799, 809, 811, 812, 818, 822, 824, 833, 836, 844, 866, 872, 888, 899, 911, 927, 933, 981, 999, 1003, 1009, 1011, 1015, 1018, 1022, 1027, 1030, 1033, 1037, 1045, 1055, 1066, 1090, 1099, 1111, 1133, 1199, 1218, 1222, 1227, 1233, 1236, 1244, 1254, 1266, 1299, 1333, 1339, 1399, 1418, 1421, 1422, 1433, 1442, 1463, 1466, 1477, 1499, 1527, 1533, 1545, 1555, 1599, 1618, 1622, 1624, 1633, 1636, 1644, 1648, 1666, 1672, 1688, 1699, 1733, 1751, 1799, 1822, 1827, 1833, 1854, 1866, 1881, 1899, 1933, 1957, 1999, 2003, 2006, 2009, 2018, 2022, 2027, 2030, 2033, 2036, 2037, 2044, 2045, 2054, 2055, 2060, 2066, 2074, 2090, 2099, 2111, 2127, 2133, 2163, 2177, 2199, 2222, 2233, 2266, 2299, 2333, 2369, 2399, 2427, 2433, 2436, 2444, 2454, 2466, 2472, 2488, 2499, 2533, 2545, 2555, 2575, 2599, 2633, 2639, 2666, 2678, 2699, 2733, 2781, 2799, 2833, 2836, 2842, 2844, 2863, 2866, 2877, 2884, 2899, 2933, 2987, 2999, 3009, 3027, 3033, 3037, 3045, 3054, 3055, 3066, 3081, 3090, 3099, 3111, 3133, 3193, 3199, 3233, 3236, 3244, 3248, 3266, 3272, 3288, 3296, 3299, 3333, 3399, 3451, 3466, 3499, 3545, 3555, 3563, 3577, 3599, 3644, 3654, 3666, 3681, 3699, 3799, 3857, 3866, 3899, 3999, 4006, 4009, 4012, 4018, 4027, 4036, 4037, 4044, 4045, 4054, 4055, 4060, 4066, 4072, 4074, 4088, 4090, 4099, 4108, 4111, 4148, 4199, 4222, 4254, 4263, 4266, 4277, 4299, 4333, 4399, 4444, 4466, 4499, 4555, 4581, 4599, 4666, 4669, 4699, 4799, 4854, 4866, 4872, 4888, 4899, 4963, 4977, 4999, 5009, 5015, 5027, 5037, 5045, 5055, 5066, 5075, 5090, 5099, 5111, 5135, 5185, 5199, 5266, 5278, 5299, 5333, 5399, 5466, 5481, 5499, 5555, 5599, 5663, 5666, 5672, 5677, 5684, 5688, 5699, 5799, 5866, 5887, 5899, 5999, 6009, 6018, 6027, 6037, 6054, 6066, 6074, 6081, 6090, 6099, 6111, 6162, 6199, 6222, 6266, 6293, 6299, 6333, 6377, 6381, 6399, 6466, 6472, 6488, 6496, 6499, 6599, 6666, 6699, 6799, 6899, 6999, 7009, 7021, 7027, 7037, 7063, 7077, 7090, 7099, 7111, 7189, 7199, 7259, 7281, 7288, 7299, 7333, 7399, 7499, 7599, 7699, 7777, 7799, 7899, 7999, 8009, 8012, 8018, 8024, 8027, 8036, 8037, 8054, 8072, 8074, 8088, 8090, 8099, 8108, 8111, 8148, 8199, 8216, 8222, 8296, 8299, 8333, 8399, 8444, 8499, 8599, 8666, 8699, 8799, 8888, 8899, 8999, 9027, 9037, 9081, 9099, 9111, 9199, 9243, 9299, 9333, 9399, 9499, 9599, 9699, 9799, 9899, 9999, 10003, 10009, 10011, 10015, 10018, 10027, 10030, 10033, 10037, 10045, 10054, 10074, 10090, 10099, 10101, 10111, 10135, 10185, 10222, 10270, 10303, 10333, 10370, 10555, 10666, 10909, 10999, 11027, 11033, 11037, 11099, 11111, 11297, 11333, 11407, 11999, 12018, 12027, 12036, 12037, 12054, 12074, 12081, 12108, 12111, 12148, 12162, 12222, 12324, 12333, 12444, 12666, 12999, 13027, 13037, 13039, 13111, 13117, 13333, 13351, 13481, 13999, 14018, 14021, 14027, 14037, 14042, 14054, 14063, 14074, 14111, 14126, 14189, 14222, 14259, 14333, 14378, 14518, 14666, 14777, 14999, 15027, 15037, 15045, 15081, 15111, 15135, 15185, 15333, 15405, 15555, 15999, 16018, 16024, 16027, 16036, 16037, 16048, 16054, 16072, 16074, 16108, 16111, 16144, 16148, 16216, 16222, 16296, 16333, 16432, 16444, 16592, 16666, 16888, 16999, 17027, 17037, 17051, 17111, 17153, 17333, 17459, 17629, 17999, 18027, 18037, 18054, 18074, 18081, 18111, 18162, 18222, 18243, 18333, 18486, 18666, 18999, 19027, 19037, 19057, 19111, 19171, 19333, 19513, 19703, 19999, 20003, 20006, 20009, 20011, 20018, 20022, 20027, 20030, 20033, 20036, 20037, 20045, 20054, 20060, 20066, 20074, 20090, 20099, 20101, 20108, 20111, 20135, 20148, 20180, 20185, 20198, 20202, 20222, 20270, 20303, 20333, 20370, 20444, 20540, 20555, 20606, 20666, 20740, 20909, 20999, 21027, 21037, 21063, 21081, 21111, 21189, 21259, 21333, 21567, 21777, 21818, 21999, 22027, 22033, 22037, 22054, 22066, 22074, 22099, 22111, 22198, 22222, 22297, 22333, 22407, 22594, 22666, 22814, 22999, 23027, 23037, 23069, 23111, 23207, 23333, 23621, 23851, 23999, 24027, 24036, 24037, 24054, 24072, 24074, 24081, 24108, 24111, 24148, 24162, 24216, 24222, 24296, 24324, 24333, 24444, 24648, 24666, 24888, 24999, 25027, 25037, 25045, 25075, 25111, 25135, 25185, 25225, 25333, 25555, 25675, 25925, 25999, 26027, 26037, 26039, 26054, 26074, 26078, 26111, 26117, 26222, 26234, 26333, 26351, 26481, 26666, 26702, 26962, 26999, 27037, 27081, 27111, 27243, 27333, 27729, 27999, 28036, 28037, 28042, 28054, 28063, 28074, 28084, 28108, 28111, 28126, 28148, 28189, 28222, 28252, 28259, 28333, 28378, 28444, 28518, 28666, 28756, 28777, 28999, 29037, 29087, 29111, 29261, 29333, 29783, 29999, 30009, 30011, 30027, 30033, 30037, 30045, 30054, 30074, 30081, 30090, 30099, 30101, 30111, 30135, 30162, 30185, 30222, 30270, 30297, 30303, 30333, 30370, 30405, 30555, 30666, 30810, 30909, 30999, 31037, 31093, 31111, 31279, 31333, 31837, 31999, 32036, 32037, 32048, 32054, 32072, 32074, 32096, 32108, 32111, 32144, 32148, 32216, 32222, 32288, 32296, 32333, 32432, 32444, 32592, 32666, 32727, 32864, 32888, 32999, 33037, 33081, 33099, 33111, 33297, 33333, 33407, 33891, 33999, 34037, 34051, 34054, 34074, 34102, 34111, 34153, 34222, 34306, 34333, 34459, 34629, 34666, 34918, 34999, 35037, 35045, 35063, 35105, 35111, 35135, 35185, 35189, 35259, 35315, 35333, 35555, 35777, 35945, 35999, 36037, 36054, 36074, 36081, 36108, 36111, 36148, 36162, 36222, 36243, 36324, 36333, 36444, 36486, 36666, 36972, 36999, 37111, 37333, 37999, 38054, 38057, 38074, 38111, 38114, 38171, 38222, 38333, 38342, 38513, 38666, 38703, 38999, 39081, 39111, 39117, 39333, 39351, 39481, 39999

Even though the first three modest numbers are prime, the prime modest numbers occupy only 0.27% of the range (108 numbers). Consecutive modest numbers are few and far between. Here are the pairs of such numbers up to 40,000:

411 and 412
811 and 812
1421 and 1422
2036 and 2037
2044 and 2045
2054 and 2055
3054 and 3055
4036 and 4037
4044 and 4045
4054 and 4055
8036 and 8037
12036 and 12037
16036 and 16037
20036 and 20037
24036 and 24037
28036 and 28037
32036 and 32037

No triplets have been found as far as I know.

It's of interest to consider the frequency of final digits of modest numbers.

Here is the breakdown of final digits in the range up to 100000 (permalink):

There are a total of 1728 modest numbers:

0 --> 67
1 --> 218
2 --> 151
3 --> 219
4 --> 161
5 --> 128
6 --> 172
7 --> 159
8 --> 144
9 --> 309

Clearly numbers ending in the digit 9 predominate. 

CONTINUED IN NEXT POST DUE TO TECHNICAL DIFFICULTIES