## Palindromes P divisible by 13 & DigitSum(P) = 2,3,…,100

Can you find palindromes divisible by 13 and whose sum of digits is:
32,   38,   39,   …,   100?

Pipo and Paul found:

32:     88088 = 2*2*2*7*11*11*13
38:     1974791 = 7*13*21701
39:     1983891 = 3*7*13*13*13*43
40:     1992991 = 7*11*11*13*181
41:     2893982 = 2*7*13*15901
42:     2976792 = 2*2*2*3*7*13*29*47
43:     2985892 = 2*2*7*13*13*631
44:     2994992 = 2*2*2*2*7*11*11*13*17
45:     3895983 = 3*3*7*13*67*71
46:     3978793 = 7*13*23*1901
47:     3987893 = 7*13*13*3371
48:     3996993 = 3*7*11*11*11*11*13
49:     4897984 = 2*2*2*2*2*2*7*13*29*29
50:     5798975 = 5*5*7*13*2549
51:     4989894 = 2*3*7*13*13*19*37
52:     4998994 = 2*7*11*11*13*227
53:     5899985 = 5*7*13*12967
54:     8984898 = 2*3*3*3*13*12799
55:     8993998 = 2*13*345923
56:     9894989 = 13*761153
57:     9977799 = 3*13*255841
58:     9986899 = 13*23*127*263
59:     9995999 = 13*768923
60:     199787991 = 3*13*149*34381
61:     199878991 = 13*29*530183
62:     199969991 = 13*83*241*769
63:     299797992 = 2*2*2*3*3*13*17*83*227
64:     299888992 = 2*2*2*2*2*13*720887
65:     299979992 = 2*2*2*13*577*4999
66:     489888984 = 2*2*2*3*13*73*137*157
67:     399898993 = 13*659*46679
68:     399989993 = 13*31*53*61*307
69:     589898985 = 3*5*13*19*113*1409
70:     589989985 = 5*13*13*31*101*223
71:     499999994 = 2*13*331*58099
72:     779989977 = 3*3*13*31*215051
73:     689999986 = 2*13*107*197*1259
74:     969989969 = 13*31*739*3257
75:     879999978 = 2*3*11*13*1025641
76:     18799899781 = 13*23*41*89*17231
77:     17899999871 = 13*13*1327*79817
78:     27899899872 = 2*2*2*2*2*3*13*67*333667
79:     19799999791 = 11*13*167*829111
80:     29799899792 = 2*2*2*2*13*13*1039*10607
81:     28899999882 = 2*3*3*3*3*13*23*617*967
82:     38899899883 = 13*59*1187*42727
83:     37999999973 = 13*2923076921
84:     47999899974 = 2*3*13*347*1773439
85:     39899999893 = 13*73*191*251*877
86:     49899899894 = 2*13*1919226919
87:     48999999984 = 2*2*2*2*3*13*521*150721
88:     58999899985 = 5*13*907690769
89:     68999799986 = 2*13*4127*643043
90:     78999699987 = 3*3*13*675211111
91:     59999999995 = 5*13*257*311*11549
92:     69999899996 = 2*2*13*1346151923
93:     79999799997 = 3*13*29*163*409*1061
94:     89999699998 = 2*13*47*61*491*2459
95:     99999599999 = 13*43*178890161
96:     2996998996992 = 2*2*2*2*2*2*2*2*2*2*3*7*11*11*13*41*2161
97:     2987999997892 = 2*2*7*13*43*190902121
98:     3978998998793 = 7*13*22003*1987241
99:     3897999997983 = 3*3*7*7*7*13*347*279919
100:     3997998997993 = 7*11*11*13*23*3229*4889

math grad - Interest: Number theory
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### 3 Responses to Palindromes P divisible by 13 & DigitSum(P) = 2,3,…,100

1. Paul says:

here are the first instances

32 , 88088
38 , 1974791
39 , 1983891
40 , 1992991
41 , 2893982
42 , 2976792
43 , 2985892
44 , 2994992
45 , 3895983
46 , 3978793
47 , 3987893
48 , 3996993
49 , 4897984
50 , 5798975
51 , 4989894
52 , 4998994
53 , 5899985
54 , 8984898
55 , 8993998
56 , 9894989
57 , 9977799
58 , 9986899
59 , 9995999
60 , 199787991
61 , 199878991
62 , 199969991
63 , 299797992
64 , 299888992
65 , 299979992
66 , 489888984
67 , 399898993
68 , 399989993
69 , 589898985
70 , 589989985
71 , 499999994
72 , 779989977
73 , 689999986
74 , 969989969
75 , 879999978
76 , 18799899781
77 , 17899999871
78 , 27899899872
79 , 19799999791
80 , 29799899792
81 , 28899999882
82 , 38899899883
83 , 37999999973
84 , 47999899974
85 , 39899999893
86 , 49899899894
87 , 48999999984
88 , 58999899985
89 , 68999799986
90 , 78999699987
91 , 59999999995
92 , 69999899996
93 , 79999799997
94 , 89999699998
95 , 99999599999
96 , 2996998996992
97 , 2987999997892
98 , 3978998998793
99 , 3897999997983
100 , 3997998997993

Paul.

2. pipo says:

Here mine, up to a digitsum of 59

2 [1001]
3 [10101]
4 [2002, 11011, 101101, 1002001]
5 [1011101]
6 [3003, 20202, 111111, 1020201]
7 [21112, 1103011]
8 [4004, 22022, 121121, 202202, 1112111, 2004002]
9 [30303, 1121211, 2013102]
10 [5005, 31213, 131131, 212212, 1130311, 1204021, 2022202]
11 [32123, 1213121, 2031302, 2105012]
12 [6006, 33033, 40404, 141141, 222222, 303303, 1222221, 2040402, 2114112, 3006003]
13 [41314, 1231321, 1305031, 2123212, 3015103]
14 [7007, 42224, 151151, 232232, 313313, 1240421, 1314131, 2132312, 2206022, 3024203, 6100016]
15 [43134, 50505, 1323231, 2141412, 2215122, 3033303, 3107013, 7001007]
16 [8008, 44044, 51415, 161161, 242242, 323323, 404404, 1332331, 1406041, 2150512, 2224222, 3042403, 3116113, 4008004, 5300035, 7010107]
17 [52325, 1341431, 1415141, 2233322, 2307032, 3051503, 3125213, 4017104, 6201026]
18 [9009, 53235, 60606, 171171, 252252, 333333, 414414, 1350531, 1424241, 2242422, 2316132, 3060603, 3134313, 3208023, 4026204, 4500054, 6210126, 7102017]
19 [54145, 61516, 1433341, 1507051, 2251522, 2325232, 3143413, 3217123, 4035304, 4109014, 5401045, 7111117, 8003008]
20 [55055, 62426, 181181, 262262, 343343, 424424, 505505, 1442441, 1516151, 2260622, 2334332, 2408042, 3152513, 3226223, 3700073, 4044404, 4118114, 5410145, 6302036, 7120217, 8012108]
21 [15951, 63336, 70707, 1451541, 1525251, 2343432, 2417142, 3161613, 3235323, 3309033, 4053504, 4127214, 4601064, 5019105, 6311136, 7203027, 8021208]
22 [16861, 64246, 71617, 191191, 272272, 353353, 434434, 515515, 1460641, 1534351, 1608061, 2352532, 2426242, 2900092, 3170713, 3244423, 3318133, 4062604, 4136314, 4610164, 5028205, 5502055, 6320236, 7212127, 8030308, 8104018]
23 [17771, 65156, 72527, 1069601, 1543451, 1617161, 2361632, 2435342, 2509052, 3253523, 3327233, 3801083, 4071704, 4145414, 4219124, 5037305, 5511155, 6403046, 7221227, 8113118, 9005009]
24 [18681, 66066, 73437, 80808, 282282, 363363, 444444, 525525, 606606, 1078701, 1552551, 1626261, 2370732, 2444442, 2518152, 3262623, 3336333, 3810183, 4080804, 4154514, 4228224, 4702074, 5046405, 5520255, 6412146, 7230327, 7304037, 8122218, 9014109]
25 [19591, 26962, 74347, 81718, 1087801, 1561651, 1635361, 1709071, 2453542, 2527252, 3271723, 3345433, 3419143, 4163614, 4237324, 4711174, 5055505, 5129215, 5603065, 6421246, 7313137, 8131318, 8205028, 9023209]
26 [27872, 75257, 82628, 292292, 373373, 454454, 535535, 616616, 1096901, 1570751, 1644461, 1718171, 2462642, 2536352, 3280823, 3354533, 3428243, 3902093, 4172714, 4246424, 4720274, 5064605, 5138315, 5612165, 6430346, 6504056, 7322237, 8140418, 8214128, 9032309, 9106019]
27 [28782, 76167, 83538, 90909, 1179711, 1653561, 1727271, 2471742, 2545452, 2619162, 3363633, 3437343, 3911193, 4181814, 4255524, 4329234, 4803084, 5073705, 5147415, 5621265, 6039306, 6513156, 7331337, 7405047, 8223228, 9041409, 9115119]
28 [29692, 77077, 84448, 91819, 383383, 464464, 545545, 626626, 707707, 1188811, 1662661, 1736371, 2480842, 2554552, 2628262, 3372733, 3446443, 3920293, 4190914, 4264624, 4338334, 4812184, 5082805, 5156515, 5630365, 5704075, 6048406, 6522256, 7340437, 7414147, 8232328, 8306038, 9050509, 9124219]
29 [37973, 85358, 92729, 1197911, 1671761, 1745471, 1819181, 2089802, 2563652, 2637362, 3381833, 3455543, 3529253, 4273724, 4347434, 4821284, 5091905, 5165615, 5239325, 5713175, 6057506, 6531356, 6605066, 7423247, 8241428, 8315138, 9133319, 9207029]
30 [38883, 86268, 93639, 393393, 474474, 555555, 636636, 717717, 1680861, 1754571, 1828281, 2098902, 2572752, 2646462, 3390933, 3464643, 3538353, 4282824, 4356534, 4830384, 4904094, 5174715, 5248425, 5722275, 6066606, 6540456, 6614166, 7432347, 7506057, 8250528, 8324238, 9142419, 9216129]
31 [39793, 87178, 94549, 1289821, 1763671, 1837381, 2581852, 2655562, 2729272, 3473743, 3547453, 4291924, 4365634, 4439344, 4913194, 5183815, 5257525, 5731375, 5805085, 6075706, 6149416, 6623266, 7441447, 7515157, 8333338, 8407048, 9151519, 9225229]
32 [88088, 95459, 484484, 565565, 646646, 727727, 808808, 1298921, 1772771, 1846481, 2590952, 2664662, 2738372, 3482843, 3556553, 4374734, 4448444, 4922294, 5192915, 5266625, 5740475, 5814185, 6084806, 6158516, 6632366, 6706076, 7450547, 7524257, 8342438, 8416148, 9160619, 9234329, 9308039]
33 [48984, 96369, 1781871, 1855581, 1929291, 2199912, 2673762, 2747472, 3491943, 3565653, 3639363, 4383834, 4457544, 4931394, 5275725, 5349435, 5823285, 6093906, 6167616, 6641466, 6715176, 7059507, 7533357, 7607067, 8351538, 8425248, 9243429, 9317139]
34 [49894, 97279, 494494, 575575, 656656, 737737, 818818, 1790971, 1864681, 1938391, 2682862, 2756572, 3574753, 3648463, 4392934, 4466644, 4940494, 5284825, 5358535, 5832385, 5906095, 6176716, 6650566, 6724276, 7068607, 7542457, 7616167, 8360638, 8434348, 8508058, 9252529, 9326239, 9800089]
35 [98189, 1399931, 1873781, 1947491, 2691962, 2765672, 2839382, 3583853, 3657563, 4475744, 4549454, 5293925, 5367635, 5841485, 5915195, 6185816, 6259526, 6733376, 6807086, 7077707, 7551557, 7625267, 8443448, 8517158, 9261629, 9335339, 9409049]
36 [99099, 585585, 666666, 747747, 828828, 909909, 1882881, 1956591, 2774772, 2848482, 3592953, 3666663, 4484844, 4558554, 5376735, 5850585, 5924295, 6194916, 6268626, 6742476, 6816186, 7086807, 7560657, 7634367, 7708077, 8452548, 8526258, 9270729, 9344439, 9418149]
37 [59995, 1891981, 1965691, 2783872, 2857582, 3675763, 3749473, 4493944, 4567654, 5385835, 5459545, 5933395, 6277726, 6751576, 6825286, 7095907, 7169617, 7643467, 7717177, 8461648, 8535358, 8609068, 9353539, 9427249, 9901099]
38 [595595, 676676, 757757, 838838, 919919, 1974791, 2792972, 2866682, 3684863, 3758573, 4576754, 5394935, 5468645, 5942495, 6286826, 6760676, 6834386, 6908096, 7178717, 7652567, 7726277, 8470748, 8544458, 8618168, 9362639, 9436349, 9910199]
39 [1983891, 2875782, 2949492, 3693963, 3767673, 4585854, 4659564, 5477745, 5951595, 6295926, 6369636, 6843486, 6917196, 7187817, 7661667, 7735377, 7809087, 8079708, 8553558, 8627268, 9371739, 9445449, 9519159]
40 [686686, 767767, 848848, 929929, 1992991, 2884882, 2958592, 3776773, 4594954, 4668664, 5486845, 5960695, 6378736, 6852586, 6926296, 7196917, 7670767, 7744477, 7818187, 8088808, 8562658, 8636368, 9380839, 9454549, 9528259]
41 [2893982, 2967692, 3785873, 3859583, 4677764, 5495945, 5569655, 6387836, 6861686, 6935396, 7279727, 7753577, 7827287, 8097908, 8571758, 8645468, 8719178, 9463649, 9537359]
42 [696696, 777777, 858858, 939939, 2976792, 3794973, 3868683, 4686864, 5578755, 6396936, 6870786, 6944496, 7288827, 7762677, 7836387, 8580858, 8654568, 8728278, 9472749, 9546459]
43 [2985892, 3877783, 4695964, 4769674, 5587855, 6479746, 6953596, 7297927, 7771777, 7845487, 7919197, 8189818, 8663668, 8737378, 9481849, 9555559, 9629269]
44 [787787, 868868, 949949, 2994992, 3886883, 4778774, 5596955, 6488846, 6962696, 7780877, 7854587, 7928297, 8198918, 8672768, 8746478, 9490949, 9564659, 9638369]
45 [3895983, 3969693, 4787874, 5679765, 6497946, 6971796, 7389837, 7863687, 7937397, 8681868, 8755578, 8829288, 9099909, 9573759, 9647469]
46 [797797, 878878, 959959, 3978793, 4796974, 5688865, 6980896, 7398937, 7872787, 7946497, 8690968, 8764678, 8838388, 9582859, 9656569]
47 [3987893, 4879784, 5697965, 6589856, 7881887, 7955597, 8299928, 8773778, 8847488, 9591959, 9665669, 9739379]
48 [888888, 969969, 3996993, 4888884, 6598956, 7890987, 7964697, 8782878, 8856588, 9674769, 9748479]
49 [4897984, 5789875, 7499947, 7973797, 8791978, 8865688, 8939398, 9683869, 9757579]
50 [898898, 979979, 5798975, 7982897, 8874788, 8948498, 9692969, 9766679]
51 [4989894, 6699966, 7991997, 8883888, 8957598, 9775779, 9849489]
52 [989989, 4998994, 8892988, 8966698, 9784879, 9858589]
53 [5899985, 8975798, 9793979, 9867689]
54 [999999, 8984898, 9876789]
55 [8993998, 9885889, 9959599]
56 [9894989, 9968699]
57 [9977799]
58 [9986899]
59 [9995999]

pipo

3. benvitalis says:

Nice work!