A number divisible by the number formed by its first and last digit. more
The first 600 gapful numbers :
100,
105,
108,
110,
120,
121,
130,
132,
135,
140,
143,
150,
154,
160,
165,
170,
176,
180,
187,
190,
192,
195,
198,
200,
220,
225,
231,
240,
242,
253,
260,
264,
275,
280,
286,
297,
300,
315,
330,
341,
352,
360,
363,
374,
385,
390,
396,
400,
405,
440,
451,
462,
473,
480,
484,
495,
500,
550,
561,
572,
583,
594,
600,
660,
671,
682,
693,
700,
770,
781,
792,
800,
880,
891,
900,
990,
1000,
1001,
1005,
1008,
1010,
1020,
1030,
1032,
1035,
1037,
1040,
1050,
1053,
1056,
1060,
1064,
1065,
1070,
1080,
1090,
1092,
1095,
1098,
1100,
1110,
1111,
1120,
1125,
1130,
1134,
1136,
1140,
1150,
1152,
1155,
1159,
1160,
1170,
1180,
1183,
1185,
1188,
1190,
1200,
1204,
1207,
1210,
1212,
1215,
1216,
1220,
1221,
1230,
1240,
1245,
1250,
1260,
1270,
1272,
1274,
1275,
1278,
1280,
1290,
1296,
1300,
1305,
1310,
1313,
1320,
1330,
1331,
1332,
1335,
1340,
1344,
1349,
1350,
1360,
1365,
1368,
1370,
1376,
1377,
1380,
1390,
1392,
1395,
1400,
1410,
1414,
1420,
1425,
1430,
1440,
1441,
1443,
1450,
1452,
1455,
1456,
1458,
1460,
1470,
1480,
1484,
1485,
1490,
1500,
1510,
1512,
1515,
1520,
1530,
1536,
1539,
1540,
1545,
1547,
1548,
1550,
1551,
1554,
1560,
1570,
1572,
1573,
1575,
1580,
1590,
1600,
1605,
1610,
1616,
1620,
1624,
1630,
1632,
1635,
1638,
1640,
1650,
1660,
1661,
1665,
1670,
1680,
1690,
1692,
1694,
1695,
1696,
1700,
1703,
1710,
1717,
1720,
1725,
1728,
1729,
1730,
1740,
1750,
1752,
1755,
1760,
1764,
1770,
1771,
1776,
1780,
1785,
1790,
1800,
1810,
1812,
1815,
1818,
1820,
1830,
1833,
1834,
1840,
1845,
1850,
1856,
1860,
1870,
1872,
1875,
1880,
1881,
1887,
1890,
1900,
1904,
1905,
1908,
1910,
1919,
1920,
1930,
1932,
1935,
1936,
1940,
1950,
1960,
1963,
1965,
1970,
1974,
1980,
1990,
1991,
1992,
1995,
1998,
2000,
2002,
2020,
2025,
2040,
2059,
2060,
2064,
2075,
2080,
2093,
2100,
2106,
2112,
2120,
2121,
2125,
2128,
2140,
2160,
2175,
2180,
2184,
2187,
2200,
2220,
2222,
2225,
2236,
2240,
2260,
2268,
2275,
2280,
2300,
2304,
2320,
2323,
2325,
2331,
2332,
2340,
2349,
2360,
2366,
2375,
2380,
2400,
2408,
2420,
2424,
2425,
2440,
2442,
2457,
2460,
2475,
2480,
2496,
2500,
2520,
2525,
2540,
2541,
2544,
2548,
2552,
2553,
2560,
2575,
2580,
2600,
2620,
2625,
2626,
2639,
2640,
2660,
2662,
2664,
2675,
2680,
2688,
2700,
2720,
2725,
2727,
2740,
2751,
2756,
2760,
2772,
2775,
2780,
2783,
2784,
2800,
2820,
2825,
2828,
2840,
2860,
2875,
2880,
2882,
2886,
2900,
2904,
2920,
2925,
2929,
2940,
2960,
2961,
2968,
2975,
2980,
2992,
2997,
3000,
3003,
3030,
3045,
3060,
3072,
3078,
3090,
3094,
3096,
3115,
3120,
3131,
3150,
3159,
3180,
3185,
3210,
3232,
3240,
3255,
3264,
3268,
3270,
3276,
3300,
3325,
3330,
3333,
3360,
3367,
3390,
3392,
3395,
3420,
3434,
3441,
3450,
3456,
3458,
3465,
3480,
3510,
3535,
3540,
3549,
3552,
3570,
3600,
3604,
3605,
3630,
3636,
3648,
3660,
3663,
3675,
3690,
3712,
3720,
3737,
3745,
3750,
3751,
3774,
3780,
3810,
3815,
3816,
3838,
3840,
3870,
3872,
3885,
3900,
3930,
3939,
3944,
3955,
3960,
3990,
3993,
3996,
4000,
4004,
4005,
4032,
4040,
4080,
4095,
4120,
4128,
4141,
4160,
4185,
4186,
4200,
4224,
4240,
4242,
4275,
4277,
4280,
4320,
4343,
4360,
4365,
4368,
4400,
4416,
4440,
4444,
4452,
4455,
4459,
4480,
4520,
4545,
4551,
4560,
4600,
4608,
4635,
4640,
4646,
4662,
4664,
4680,
4720,
4725,
4747,
4760,
4773,
4800,
4815,
4840,
4848,
4872,
4876,
4880,
4884,
4905,
4920,
4949,
4960,
4961,
4995,
5000,
5005,
5050,
5096,
5100,
5115,
5150,
5151,
5184,
5187,
5200,
5225,
5250,
5252,
5278,
5300,
5335,
5350,
5353,
5369,
5376,
5400,
5445,
5450,
5454,
5500,
5512,
5550,
5555,
5568,
5600,
5650,
5656,
5661,
5665,
5700,
5724,
5750,
5757,
5772.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 74623687 values, from 100 to 1000000000).
n\r | 0 | 1 |
2 | 51767230 | 22856457 | 2 |
3 | 40032989 | 17295361 | 17295337 | 3 |
4 | 36835353 | 11428222 | 14931877 | 11428235 | 4 |
5 | 44024729 | 7844198 | 8041629 | 7274568 | 7438563 | 5 |
6 | 26755478 | 4789475 | 12505866 | 13277511 | 12505886 | 4789471 | 6 |
7 | 17664603 | 9493175 | 9493188 | 9493174 | 9493170 | 9493181 | 9493196 | 7 |
8 | 23005482 | 5714108 | 7465934 | 5714127 | 13829871 | 5714114 | 7465943 | 5714108 | 8 |
9 | 17263286 | 5765136 | 5765114 | 11384842 | 5765103 | 5765114 | 11384861 | 5765122 | 5765109 | 9 |
10 | 31432997 | 2984424 | 5692469 | 2724461 | 5231883 | 12591732 | 4859774 | 2349160 | 4550107 | 2206680 | 10 |
11 | 11072961 | 6355062 | 6355046 | 6355079 | 6355059 | 6355074 | 6355079 | 6355079 | 6355082 | 6355090 | 6355076 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.