## Palindromes P divisible by 7 & DigitSum(P) = 2,3,…,40

Can you find palindromes divisible by 7 and whose sum of digits is:

5, 27, 29, 31, 33, 35, 37, 38, 39, 40?

Pipo found :

4 :   1002001

5 :   1011101

6 :   20202, 1020201

7 :   21112, 1103011

8 :   22022, 202202, 1112111, 2004002, 3100013

9 :   30303, 1121211, 2013102, 4001004

10 :   31213, 212212, 1130311, 1204021, 2022202, 2300032, 4010104

11 :   32123, 1009001, 1213121, 2031302, 2105012, 3201023

12 :   33033, 40404, 222222, 303303, 1018101, 1222221, 1500051, 2040402, 2114112, 3006003, 3210123, 4102014

13 :   20902 , 41314, 1027201, 1231321, 1305031, 2123212, 2401042, 3015103, 4111114, 5003005

14 :   21812, 42224, 70007, 232232, 313313, 700007, 1036301, 1240421, 1314131, 2132312, 2206022, 2410142, 3024203, 3302033, 4120214, 5012105, 7000007

15 :   22722, 43134, 50505, 1045401, 1119111, 1323231, 1601061, 2141412, 2215122, 3033303, 3107013, 3311133, 4203024, 5021205

16 :   23632, 44044, 51415, 242242, 323323, 404404, 710017, 1054501, 1128211, 1332331, 1406041, 1610161, 2150512, 2224222, 2502052, 3042403, 3116113, 3320233, 4008004, 4212124, 5030305, 5104015, 6200026

17 :   24542, 31913, 52325, 80108, 1063601, 1137311, 1341431, 1415141, 2029202, 2233322, 2307032, 2511152, 3051503, 3125213, 3403043, 4017104, 4221224, 5113115, 6005006, 7101017

18 :   25452, 32823, 53235, 60606, 81018, 252252, 333333, 414414, 720027, 801108, 1072701, 1146411, 1350531, 1424241, 1702071, 2038302, 2242422, 2316132, 2520252, 3060603, 3134313, 3208023, 3412143, 4026204, 4230324, 4304034, 5122215, 5400045, 6014106, 7110117, 8002008

19 :   26362, 33733, 54145, 61516, 1081801, 1155511, 1229221, 1433341, 1507051, 1711171, 2047402, 2251522, 2325232, 2603062, 3143413, 3217123, 3421243, 4035304, 4109014, 4313134, 5131315, 5205025, 6023206, 6301036, 8011108

20 :   27272, 34643, 55055, 62426, 90209, 262262, 343343, 424424, 505505, 730037, 811118, 1090901, 1164611, 1238321, 1442441, 1516151, 1720271, 2056502, 2260622, 2334332, 2408042, 2612162, 3152513, 3226223, 3430343, 3504053, 4044404, 4118114, 4322234, 4600064, 5140415, 5214125, 6032306, 6106016, 6310136, 7202027, 8020208

Paul found :

41 :   97979
42 :   98889
43 :   99799
44 :   1898981
45 :   2799972
46 :   3978793
47 :   1999991
48 :   3996993
49 :   4897984
50 :   5798975
51 :   4989894
52 :   4998994
53 :   5899985
54 :   7896987
55 :   7979797
56 :   7988897
57 :   7997997
58 :   8898988
59 :   9799979
60 :   179989971
61 :   8999998
62 :   299787992
63 :   279999972
64 :   299969992
65 :   399797993
66 :   399888993
67 :   399979993
68 :   589888985
69 :   499898994
70 :   499989994
71 :   689898986
72 :   689989986
73 :   599999995
74 :   879989978
75 :   789999987
76 :   999787999
77 :   979999979
78 :   999969999
79 :   18899999881
80 :   28899899882
81 :   27999999972
82 :   37999899973
83 :   29899999892
84 :   39899899893
85 :   38999999983
86 :   48999899984
87 :   58999799985
88 :   68999699986
89 :   49999999994
90 :   59999899995
91 :   69999799996
92 :   79999699997
93 :   88899999888
94 :   98899899889
95 :   97999999979
96 :   1898998998981
97 :   99899999899
98 :   1989998999891
99 :   2799999999972
100 :   2899998999982

math grad - Interest: Number theory
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### 3 Responses to Palindromes P divisible by 7 & DigitSum(P) = 2,3,…,40

1. pipo says:

Here are some more:
2 [1001]
3 [10101]
4 [2002, 11011, 101101, 1002001]
5 [1011101]
6 [3003, 20202, 111111, 1020201]
7 [7, 21112, 1103011]
8 [161, 4004, 22022, 121121, 202202, 1112111, 2004002, 3100013]
9 [252, 30303, 1121211, 2013102, 4001004]
10 [343, 5005, 10801, 31213, 131131, 212212, 1130311, 1204021, 2022202, 2300032, 4010104]
11 [434, 11711, 32123, 1009001, 1213121, 2031302, 2105012, 3201023]
12 [525, 6006, 12621, 33033, 40404, 141141, 222222, 303303, 1018101, 1222221, 1500051, 2040402, 2114112, 3006003, 3210123, 4102014]
13 [616, 13531, 20902, 41314, 1027201, 1231321, 1305031, 2123212, 2401042, 3015103, 4111114, 5003005]
14 [77, 707, 7007, 14441, 21812, 42224, 70007, 151151, 232232, 313313, 700007, 1036301, 1240421, 1314131, 2132312, 2206022, 2410142, 3024203, 3302033, 4120214, 5012105, 7000007]
15 [15351, 22722, 43134, 50505, 1045401, 1119111, 1323231, 1601061, 2141412, 2215122, 3033303, 3107013, 3311133, 4203024, 5021205]
16 [1771, 8008, 16261, 23632, 44044, 51415, 161161, 242242, 323323, 404404, 710017, 1054501, 1128211, 1332331, 1406041, 1610161, 2150512, 2224222, 2502052, 3042403, 3116113, 3320233, 4008004, 4212124, 5030305, 5104015, 6200026]
17 [17171, 24542, 31913, 52325, 80108, 1063601, 1137311, 1341431, 1415141, 2029202, 2233322, 2307032, 2511152, 3051503, 3125213, 3403043, 4017104, 4221224, 5113115, 6005006, 7101017]
18 [2772, 9009, 18081, 25452, 32823, 53235, 60606, 81018, 108801, 171171, 252252, 333333, 414414, 720027, 801108, 1072701, 1146411, 1350531, 1424241, 1702071, 2038302, 2242422, 2316132, 2520252, 3060603, 3134313, 3208023, 3412143, 4026204, 4230324, 4304034, 5122215, 5400045, 6014106, 7110117, 8002008]
19 [595, 26362, 33733, 54145, 61516, 1081801, 1155511, 1229221, 1433341, 1507051, 1711171, 2047402, 2251522, 2325232, 2603062, 3143413, 3217123, 3421243, 4035304, 4109014, 4313134, 5131315, 5205025, 6023206, 6301036, 8011108]
20 [686, 3773, 27272, 34643, 55055, 62426, 90209, 118811, 181181, 262262, 343343, 424424, 505505, 730037, 811118, 1090901, 1164611, 1238321, 1442441, 1516151, 1720271, 2056502, 2260622, 2334332, 2408042, 2612162, 3152513, 3226223, 3430343, 3504053, 4044404, 4118114, 4322234, 4600064, 5140415, 5214125, 6032306, 6106016, 6310136, 7202027, 8020208]
21 [777, 28182, 35553, 42924, 63336, 70707, 91119, 1173711, 1247421, 1451541, 1525251, 1803081, 2065602, 2139312, 2343432, 2417142, 2621262, 3161613, 3235323, 3309033, 3513153, 4053504, 4127214, 4331334, 4405044, 5019105, 5223225, 5501055, 6041406, 6115116, 7007007, 7211127, 8103018]
22 [868, 4774, 29092, 36463, 43834, 64246, 71617, 92029, 128821, 191191, 209902, 272272, 353353, 434434, 515515, 740047, 821128, 902209, 1182811, 1256521, 1460641, 1534351, 1608061, 1812181, 2074702, 2148412, 2352532, 2426242, 2630362, 2704072, 3170713, 3244423, 3318133, 3522253, 3800083, 4062604, 4136314, 4340434, 4414144, 5028205, 5232325, 5306035, 5510155, 6050506, 6124216, 6402046, 7016107, 7220227, 8112118, 9004009]
23 [959, 16961, 37373, 44744, 65156, 72527, 1191911, 1265621, 1339331, 1543451, 1617161, 1821281, 2083802, 2157512, 2361632, 2435342, 2509052, 2713172, 3049403, 3253523, 3327233, 3531353, 3605063, 4071704, 4145414, 4219124, 4423244, 4701074, 5037305, 5241425, 5315135, 6133316, 6207026, 6411146, 7025207, 7303037, 8121218, 9013109]
24 [5775, 17871, 38283, 45654, 66066, 73437, 80808, 138831, 219912, 282282, 363363, 444444, 525525, 606606, 750057, 831138, 912219, 1274721, 1348431, 1552551, 1626261, 1830381, 1904091, 2092902, 2166612, 2370732, 2444442, 2518152, 2722272, 3058503, 3262623, 3336333, 3540453, 3614163, 4080804, 4154514, 4228224, 4432344, 4506054, 4710174, 5046405, 5250525, 5324235, 5602065, 6142416, 6216126, 6420246, 7034307, 7108017, 7312137, 8130318, 8204028, 9022209, 9300039]
25 [18781, 39193, 46564, 53935, 74347, 81718, 1079701, 1283821, 1357531, 1561651, 1635361, 1709071, 1913191, 2175712, 2249422, 2453542, 2527252, 2731372, 2805082, 3067603, 3271723, 3345433, 3419143, 3623263, 3901093, 4163614, 4237324, 4441444, 4515154, 5055505, 5129215, 5333335, 5407045, 5611165, 6151516, 6225226, 6503056, 7043407, 7117117, 7321237, 8009008, 8213128, 9031309, 9105019]
26 [6776, 19691, 47474, 54845, 75257, 82628, 148841, 229922, 292292, 373373, 454454, 535535, 616616, 760067, 841148, 922229, 1088801, 1292921, 1366631, 1570751, 1644461, 1718171, 1922291, 2184812, 2258522, 2462642, 2536352, 2740472, 2814182, 3076703, 3280823, 3354533, 3428243, 3632363, 3706073, 3910193, 4172714, 4246424, 4450544, 4524254, 4802084, 5064605, 5138315, 5342435, 5416145, 5620265, 6160616, 6234326, 6308036, 6512156, 7052507, 7126217, 7330337, 7404047, 8018108, 8222228, 8500058, 9040409, 9114119]
27 [27972, 48384, 55755, 76167, 83538, 90909, 1097901, 1375731, 1449441, 1653561, 1727271, 1931391, 2193912, 2267622, 2471742, 2545452, 2619162, 2823282, 3085803, 3159513, 3363633, 3437343, 3641463, 3715173, 4181814, 4255524, 4329234, 4533354, 4607064, 4811184, 5073705, 5147415, 5351535, 5425245, 5703075, 6039306, 6243426, 6317136, 6521256, 7061607, 7135317, 7209027, 7413147, 8027208, 8231328, 8305038, 9123219, 9401049]
28 [7777, 28882, 49294, 56665, 77077, 84448, 91819, 158851, 239932, 383383, 464464, 545545, 626626, 707707, 770077, 851158, 932239, 1384831, 1458541, 1662661, 1736371, 1940491, 2276722, 2480842, 2554552, 2628262, 2832382, 2906092, 3094903, 3168613, 3372733, 3446443, 3650563, 3724273, 4190914, 4264624, 4338334, 4542454, 4616164, 4820284, 5082805, 5156515, 5360635, 5434345, 5508055, 5712175, 6048406, 6252526, 6326236, 6530356, 6604066, 7070707, 7144417, 7218127, 7422247, 7700077, 8036308, 8240428, 8314138, 9132319, 9206029, 9410149]
29 [29792, 57575, 64946, 85358, 92729, 1189811, 1393931, 1467641, 1671761, 1745471, 1819181, 2285822, 2359532, 2563652, 2637362, 2841482, 2915192, 3177713, 3381833, 3455543, 3529253, 3733373, 3807083, 4069604, 4273724, 4347434, 4551554, 4625264, 4903094, 5091905, 5165615, 5239325, 5443445, 5517155, 5721275, 6057506, 6261626, 6335336, 6409046, 6613166, 7153517, 7227227, 7431347, 7505057, 8045408, 8119118, 8323238, 8601068, 9141419, 9215129]
30 [8778, 58485, 65856, 86268, 93639, 168861, 249942, 393393, 474474, 555555, 636636, 717717, 780087, 861168, 942249, 1198911, 1476741, 1680861, 1754571, 1828281, 2294922, 2368632, 2572752, 2646462, 2850582, 2924292, 3186813, 3390933, 3464643, 3538353, 3742473, 3816183, 4078704, 4282824, 4356534, 4560654, 4634364, 4708074, 4912194, 5174715, 5248425, 5452545, 5526255, 5730375, 5804085, 6066606, 6270726, 6344436, 6418146, 6622266, 6900096, 7162617, 7236327, 7440447, 7514157, 8054508, 8128218, 8332338, 8406048, 8610168, 9150519, 9224229, 9502059]
31 [38983, 59395, 66766, 87178, 94549, 1485841, 1559551, 1763671, 1837381, 2099902, 2377732, 2581852, 2655562, 2729272, 2933392, 3195913, 3269623, 3473743, 3547453, 3751573, 3825283, 4087804, 4291924, 4365634, 4439344, 4643464, 4717174, 4921294, 5183815, 5257525, 5461645, 5535355, 5609065, 5813185, 6075706, 6149416, 6353536, 6427246, 6631366, 6705076, 7171717, 7245427, 7319137, 7523257, 7801087, 8063608, 8137318, 8341438, 8415148, 9029209, 9233329, 9307039, 9511159]
32 [9779, 39893, 67676, 88088, 95459, 178871, 259952, 484484, 565565, 646646, 727727, 790097, 808808, 871178, 952259, 1494941, 1568651, 1772771, 1846481, 2386832, 2590952, 2664662, 2738372, 2942492, 3278723, 3482843, 3556553, 3760673, 3834383, 3908093, 4096904, 4374734, 4448444, 4652564, 4726274, 4930394, 5192915, 5266625, 5470745, 5544455, 5618165, 5822285, 6084806, 6158516, 6362636, 6436346, 6640466, 6714176, 7180817, 7254527, 7328237, 7532357, 7606067, 7810187, 8072708, 8146418, 8350538, 8424248, 8702078, 9038309, 9242429, 9316139, 9520259]
33 [68586, 75957, 96369, 1299921, 1577751, 1781871, 1855581, 1929291, 2395932, 2469642, 2673762, 2747472, 2951592, 3287823, 3491943, 3565653, 3639363, 3843483, 3917193, 4179714, 4383834, 4457544, 4661664, 4735374, 4809084, 5275725, 5349435, 5553555, 5627265, 5831385, 5905095, 6093906, 6167616, 6371736, 6445446, 6519156, 6723276, 7059507, 7263627, 7337337, 7541457, 7615167, 8081808, 8155518, 8229228, 8433348, 8507058, 8711178, 9047409, 9251529, 9325239, 9603069]
34 [69496, 76867, 97279, 188881, 269962, 494494, 575575, 656656, 737737, 818818, 881188, 962269, 1586851, 1790971, 1864681, 1938391, 2478742, 2682862, 2756572, 2960692, 3296923, 3574753, 3648463, 3852583, 3926293, 4188814, 4392934, 4466644, 4670764, 4744474, 4818184, 5284825, 5358535, 5562655, 5636365, 5840485, 5914195, 6176716, 6380836, 6454546, 6528256, 6732376, 6806086, 7068607, 7272727, 7346437, 7550557, 7624267, 7902097, 8090908, 8164618, 8238328, 8442448, 8516158, 8720278, 9056509, 9260629, 9334339, 9408049, 9612169]
35 [49994, 77777, 98189, 1595951, 1669661, 1873781, 1947491, 2487842, 2691962, 2765672, 2839382, 3379733, 3583853, 3657563, 3861683, 3935393, 4197914, 4475744, 4549454, 4753574, 4827284, 5089805, 5293925, 5367635, 5571755, 5645465, 5719175, 5923295, 6185816, 6259526, 6463646, 6537356, 6741476, 6815186, 7077707, 7281827, 7355537, 7429247, 7633367, 7707077, 7911197, 8173718, 8247428, 8451548, 8525258, 8803088, 9065609, 9139319, 9343439, 9417149, 9621269]
36 [78687, 99099, 198891, 279972, 585585, 666666, 747747, 828828, 891198, 909909, 972279, 1678761, 1882881, 1956591, 2496942, 2774772, 2848482, 3388833, 3592953, 3666663, 3870783, 3944493, 4484844, 4558554, 4762674, 4836384, 5098905, 5376735, 5580855, 5654565, 5728275, 5932395, 6194916, 6268626, 6472746, 6546456, 6750576, 6824286, 7086807, 7290927, 7364637, 7438347, 7642467, 7716177, 7920297, 8182818, 8256528, 8460648, 8534358, 8608068, 8812188, 9074709, 9148419, 9352539, 9426249, 9630369, 9704079]
37 [79597, 86968, 1687861, 1891981, 1965691, 2579752, 2783872, 2857582, 3397933, 3675763, 3749473, 3953593, 4289824, 4493944, 4567654, 4771774, 4845484, 4919194, 5385835, 5459545, 5663665, 5737375, 5941495, 6277726, 6481846, 6555556, 6629266, 6833386, 6907096, 7095907, 7169617, 7373737, 7447447, 7651567, 7725277, 8191918, 8265628, 8339338, 8543458, 8617168, 8821288, 9083809, 9157519, 9361639, 9435349, 9509059, 9713179]
38 [87878, 289982, 595595, 676676, 757757, 838838, 919919, 982289, 1696961, 1974791, 2588852, 2792972, 2866682, 3684863, 3758573, 3962693, 4298924, 4576754, 4780874, 4854584, 4928294, 5394935, 5468645, 5672765, 5746475, 5950595, 6286826, 6490946, 6564656, 6638366, 6842486, 6916196, 7178717, 7382837, 7456547, 7660667, 7734377, 7808087, 8274728, 8348438, 8552558, 8626268, 8830388, 8904098, 9092909, 9166619, 9370739, 9444449, 9518159, 9722279]
39 [88788, 1779771, 1983891, 2597952, 2875782, 2949492, 3489843, 3693963, 3767673, 3971793, 4585854, 4659564, 4863684, 4937394, 5199915, 5477745, 5681865, 5755575, 5829285, 6295926, 6369636, 6573756, 6647466, 6851586, 6925296, 7187817, 7391937, 7465647, 7539357, 7743477, 7817187, 8079708, 8283828, 8357538, 8561658, 8635368, 8709078, 8913198, 9175719, 9249429, 9453549, 9527259, 9731379, 9805089]
40 [89698, 299992, 686686, 767767, 848848, 929929, 992299, 1788871, 1992991, 2884882, 2958592, 3498943, 3776773, 3980893, 4594954, 4668664, 4872784, 4946494, 5486845, 5690965, 5764675, 5838385, 6378736, 6582856, 6656566, 6860686, 6934396, 7196917, 7474747, 7548457, 7752577, 7826287, 8088808, 8292928, 8366638, 8570758, 8644468, 8718178, 8922298, 9184819, 9258529, 9462649, 9536359, 9740479, 9814189]

pipo

2. Paul says:

Here are the first instances when the sum is equal to 41 to 100

41 , 97979
42 , 98889
43 , 99799
44 , 1898981
45 , 2799972
46 , 3978793
47 , 1999991
48 , 3996993
49 , 4897984
50 , 5798975
51 , 4989894
52 , 4998994
53 , 5899985
54 , 7896987
55 , 7979797
56 , 7988897
57 , 7997997
58 , 8898988
59 , 9799979
60 , 179989971
61 , 8999998
62 , 299787992
63 , 279999972
64 , 299969992
65 , 399797993
66 , 399888993
67 , 399979993
68 , 589888985
69 , 499898994
70 , 499989994
71 , 689898986
72 , 689989986
73 , 599999995
74 , 879989978
75 , 789999987
76 , 999787999
77 , 979999979
78 , 999969999
79 , 18899999881
80 , 28899899882
81 , 27999999972
82 , 37999899973
83 , 29899999892
84 , 39899899893
85 , 38999999983
86 , 48999899984
87 , 58999799985
88 , 68999699986
89 , 49999999994
90 , 59999899995
91 , 69999799996
92 , 79999699997
93 , 88899999888
94 , 98899899889
95 , 97999999979
96 , 1898998998981
97 , 99899999899
98 , 1989998999891
99 , 2799999999972
100 , 2899998999982

Paul

3. benvitalis says:

Nice job, Pipo and Paul!