What are the numbers divisible by 111?
111, 222, 333, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443, 1554, 1665, 1776, 1887, 1998, 2109, 2220, 2331, 2442, 2553, 2664, 2775, 2886, 2997, 3108, 3219, 3330, 3441, 3552, 3663, 3774, 3885, 3996, 4107, 4218, 4329, 4440, 4551, 4662, 4773, 4884, 4995, 5106, 5217, 5328, 5439, 5550, 5661, 5772, 5883, 5994, 6105, 6216, 6327, 6438, 6549, 6660, 6771, 6882, 6993, 7104, 7215, 7326, 7437, 7548, 7659, 7770, 7881, 7992, 8103, 8214, 8325, 8436, 8547, 8658, 8769, 8880, 8991, 9102, 9213, 9324, 9435, 9546, 9657, 9768, 9879, 9990, 10101, 10212, 10323, 10434, 10545, 10656, 10767, 10878, 10989, 11100, 11211, 11322, 11433, 11544, 11655, 11766, 11877, 11988, 12099, 12210, 12321, 12432, 12543, 12654, 12765, 12876, 12987, 13098, 13209, 13320, 13431, 13542, 13653, 13764, 13875, 13986, 14097, 14208, 14319, 14430, 14541, 14652, 14763, 14874, 14985, 15096, 15207, 15318, 15429, 15540, 15651, 15762, 15873, 15984, 16095, 16206, 16317, 16428, 16539, 16650, 16761, 16872, 16983, 17094, 17205, 17316, 17427, 17538, 17649, 17760, 17871, 17982, 18093, 18204, 18315, 18426, 18537, 18648, 18759, 18870, 18981, 19092, 19203, 19314, 19425, 19536, 19647, 19758, 19869, 19980, 20091, 20202, 20313, 20424, 20535, 20646, 20757, 20868, 20979, 21090, 21201, 21312, 21423, 21534, 21645, 21756, 21867, 21978, 22089, 22200, 22311, 22422, 22533, 22644, 22755, 22866, 22977, 23088, 23199, 23310, 23421, 23532, 23643, 23754, 23865, 23976, 24087, 24198, 24309, 24420, 24531, 24642, 24753, 24864, 24975, 25086, 25197, 25308, 25419, 25530, 25641, 25752, 25863, 25974, 26085, 26196, 26307, 26418, 26529, 26640, 26751, 26862, 26973, 27084, 27195, 27306, 27417, 27528, 27639, 27750, 27861, 27972, 28083, 28194, 28305, 28416, 28527, 28638, 28749, 28860, 28971, 29082, 29193, 29304, 29415, 29526, 29637, 29748, 29859, 29970, 30081, 30192, 30303, 30414, 30525, 30636, 30747, 30858, 30969, 31080, 31191, 31302, 31413, 31524, 31635, 31746, 31857, 31968, 32079, 32190, 32301, 32412, 32523, 32634, 32745, 32856, 32967, 33078, 33189, 33300, 33411, 33522, 33633, 33744, 33855, 33966, 34077, 34188, 34299, 34410, 34521, 34632, 34743, 34854, 34965, 35076, 35187, 35298, 35409, 35520, 35631, 35742, 35853, 35964, 36075, 36186, 36297, 36408, 36519, 36630, 36741, 36852, 36963, 37074, 37185, 37296, 37407, 37518, 37629, 37740, 37851, 37962, 38073, 38184, 38295, 38406, 38517, 38628, 38739, 38850, 38961, 39072, 39183, 39294, 39405, 39516, 39627, 39738, 39849, 39960, 40071, 40182, 40293, 40404, 40515, 40626, 40737, 40848, 40959, 41070, 41181, 41292, 41403, 41514, 41625, 41736, 41847, 41958, 42069, 42180, 42291, 42402, 42513, 42624, 42735, 42846, 42957, 43068, 43179, 43290, 43401, 43512, 43623, 43734, 43845, 43956, 44067, 44178, 44289, 44400, 44511, 44622, 44733, 44844, 44955, 45066, 45177, 45288, 45399, 45510, 45621, 45732, 45843, 45954, 46065, 46176, 46287, 46398, 46509, 46620, 46731, 46842, 46953, 47064, 47175, 47286, 47397, 47508, 47619, 47730, 47841, 47952, 48063, 48174, 48285, 48396, 48507, 48618, 48729, 48840, 48951, 49062, 49173, 49284, 49395, 49506, 49617, 49728, 49839, 49950, 50061, 50172, 50283, 50394, 50505, 50616, 50727, 50838, 50949, 51060, 51171, 51282, 51393, 51504, 51615, 51726, 51837, 51948, 52059, 52170, 52281, 52392, 52503, 52614, 52725, 52836, 52947, 53058, 53169, 53280, 53391, 53502, 53613, 53724, 53835, 53946, 54057, 54168, 54279, 54390, 54501, 54612, 54723, 54834, 54945, 55056, 55167, 55278, 55389, 55500, 55611, 55722, 55833, 55944, 56055, 56166, 56277, 56388, 56499, 56610, 56721, 56832, 56943, 57054, 57165, 57276, 57387, 57498, 57609, 57720, 57831, 57942, 58053, 58164, 58275, 58386, 58497, 58608, 58719, 58830, 58941, 59052, 59163, 59274, 59385, 59496, 59607, 59718, 59829, 59940, 60051, 60162, 60273, 60384, 60495, 60606, 60717, 60828, 60939, 61050, 61161, 61272, 61383, 61494, 61605, 61716, 61827, 61938, 62049, 62160, 62271, 62382, 62493, 62604, 62715, 62826, 62937, 63048, 63159, 63270, 63381, 63492, 63603, 63714, 63825, 63936, 64047, 64158, 64269, 64380, 64491, 64602, 64713, 64824, 64935, 65046, 65157, 65268, 65379, 65490, 65601, 65712, 65823, 65934, 66045, 66156, 66267, 66378, 66489, 66600, 66711, 66822, 66933, 67044, 67155, 67266, 67377, 67488, 67599, 67710, 67821, 67932, 68043, 68154, 68265, 68376, 68487, 68598, 68709, 68820, 68931, 69042, 69153, 69264, 69375, 69486, 69597, 69708, 69819, 69930, 70041, 70152, 70263, 70374, 70485, 70596, 70707, 70818, 70929, 71040, 71151, 71262, 71373, 71484, 71595, 71706, 71817, 71928, 72039, 72150, 72261, 72372, 72483, 72594, 72705, 72816, 72927, 73038, 73149, 73260, 73371, 73482, 73593, 73704, 73815, 73926, 74037, 74148, 74259, 74370, 74481, 74592, 74703, 74814, 74925, 75036, 75147, 75258, 75369, 75480, 75591, 75702, 75813, 75924, 76035, 76146, 76257, 76368, 76479, 76590, 76701, 76812, 76923, 77034, 77145, 77256, 77367, 77478, 77589, 77700, 77811, 77922, 78033, 78144, 78255, 78366, 78477, 78588, 78699, 78810, 78921, 79032, 79143, 79254, 79365, 79476, 79587, 79698, 79809, 79920, 80031, 80142, 80253, 80364, 80475, 80586, 80697, 80808, 80919, 81030, 81141, 81252, 81363, 81474, 81585, 81696, 81807, 81918, 82029, 82140, 82251, 82362, 82473, 82584, 82695, 82806, 82917, 83028, 83139, 83250, 83361, 83472, 83583, 83694, 83805, 83916, 84027, 84138, 84249, 84360, 84471, 84582, 84693, 84804, 84915, 85026, 85137, 85248, 85359, 85470, 85581, 85692, 85803, 85914, 86025, 86136, 86247, 86358, 86469, 86580, 86691, 86802, 86913, 87024, 87135, 87246, 87357, 87468, 87579, 87690, 87801, 87912, 88023, 88134, 88245, 88356, 88467, 88578, 88689, 88800, 88911, 89022, 89133, 89244, 89355, 89466, 89577, 89688, 89799, 89910, 90021, 90132, 90243, 90354, 90465, 90576, 90687, 90798, 90909, 91020, 91131, 91242, 91353, 91464, 91575, 91686, 91797, 91908, 92019, 92130, 92241, 92352, 92463, 92574, 92685, 92796, 92907, 93018, 93129, 93240, 93351, 93462, 93573, 93684, 93795, 93906, 94017, 94128, 94239, 94350, 94461, 94572, 94683, 94794, 94905, 95016, 95127, 95238, 95349, 95460, 95571, 95682, 95793, 95904, 96015, 96126, 96237, 96348, 96459, 96570, 96681, 96792, 96903, 97014, 97125, 97236, 97347, 97458, 97569, 97680, 97791, 97902, 98013, 98124, 98235, 98346, 98457, 98568, 98679, 98790, 98901, 99012, 99123, 99234, 99345, 99456, 99567, 99678, 99789, 99900
- There is a total of 900 numbers (up to 100000) that are divisible by 111.
- The sum of these numbers is 45004950.
- The arithmetic mean of these numbers is 50005.5.
How to find the numbers divisible by 111?
Finding all the numbers that can be divided by 111 is essentially the same as searching for the multiples of 111: if a number N is a multiple of 111, then 111 is a divisor of N.
Indeed, if we assume that N is a multiple of 111, this means there exists an integer k such that:
Conversely, the result of N divided by 111 is this same integer k (without any remainder):
From this we can see that, theoretically, there's an infinite quantity of multiples of 111 (we can keep multiplying it by increasingly larger integers, without ever reaching the end).
However, in this instance, we've chosen to set an arbitrary limit (specifically, the multiples of 111 less than 100000):
- 1 × 111 = 111
- 2 × 111 = 222
- 3 × 111 = 333
- ...
- 899 × 111 = 99789
- 900 × 111 = 99900