題目：X mod f(x)

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題意：問在區(qū)間[A,B]之間，有多少個數滿足，x%f(x)=0，f(x)代表x的各位數字之和。

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解析：本題一是可以通過打表來計算，二是可以通過數位DP來求解。

我們先來說打表吧。。。

對于打表，當然我們不能直接打，這里有技巧。我們可以這樣：每隔100000我們統計一次，這樣在10^9范圍內，我們得到10000個數。

然后就很好解決了。。。

#include <iostream> #include <string.h> #include <stdio.h>using namespace std;int f(int x) {if(x==0) return 0;return f(x/10)+x%10; }int t,a,b;int 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4,4857,4715,4646,4545,4440,4390,4304,4207,7896,7662,7512,7221,6927,6765,6570,6329,6128,5961,7641,7419,7264,7008,6759,6573,6383,6207,5955,5812,7377,7165,7051,6826,6571,6402,6243,6070,5809,5673,7122,6928,6836,6609,6382,6253,6079,5930,5684,5563,6893,6688,6608,6388,6220,6058,5905,5797,5572,5455,6671,6415,6389,6213,6027,5894,5760,5660,5438,5340,6457,6215,6167,6015,5855,5741,5629,5494,5356,5262,6279,6037,5995,5837,5722,5603,5487,5378,5242,5165,6109,5859,5817,5721,5602,5452,5367,5287,5140,5047,5929,5717,5700,5591,5465,5341,5271,5172,5041,4984,7672,7503,7215,6981,6695,6534,6341,6160,5938,5773,7410,7273,7003,6787,6524,6375,6194,6003,5789,5671,7174,7054,6794,6579,6373,6213,6049,5883,5680,5543,6951,6822,6589,6413,6218,6036,5902,5779,5543,5431,6705,6589,6391,6216,6029,5893,5782,5647,5431,5341,6470,6357,6182,6008,5867,5741,5619,5508,5344,5227,6292,6136,5989,5858,5742,5586,5498,5400,5218,5112,6106,5942,5829,5729,5593,5469,5389,5288,5134,5024,5905,5794,5700,5577,5467,5366,5267,5150,5048,4951,5786,5652,5539,5459,5388,5237,5154,5077,4973,4854,7500,7220,6976,6727,6481,6349,6136,5968,5792,5637,7284,6994,6775,6561,6312,6170,5981,5847,5655,5500,7065,6778,6602,6412,6170,6019,5890,5744,5503,5404,6838,6590,6408,6222,6001,5901,5761,5618,5432,5310,6621,6381,6187,6040,5883,5733,5606,5508,5339,5212,6407,6151,5997,5870,5730,5599,5491,5393,5192,5100,6206,5962,5848,5716,5573,5495,5379,5271,5113,5026,6028,5826,5687,5594,5494,5375,5248,5176,5036,4931,5865,5634,5562,5467,5372,5248,5162,5075,4935,4833,5702,5518,5461,5361,5243,5158,5080,4978,4854,4767,7222,6969,6768,6534,6290,6127,5965,5810,5615,5483,6993,6768,6582,6364,6120,5995,5830,5664,5496,5374,6808,6594,6392,6193,6002,5869,5709,5557,5417,5260,6627,6375,6225,6079,5850,5720,5613,5485,5293,5161,6413,6176,6036,5887,5705,5602,5495,5363,5194,5101,6226,5998,5848,5721,5582,5487,5366,5257,5102,4983,6056,5823,5704,5594,5470,5334,5262,5165,4993,4885,5890,5661,5582,5479,5361,5247,5173,5074,4921,4825,5736,5556,5438,5352,5274,5163,5040,4959,4832,4721,5598,5425,5329,5245,5183,5062,4961,4881,4756,4619,6946,6788,6541,6349,6104,5946,5794,5646,5494,5427,6771,6578,6359,6196,5970,5818,5663,5555,5361,5303,6579,6380,6223,6038,5820,5716,5562,5441,5259,5204,6396,6212,6042,5885,5706,5602,5452,5338,5185,5124,6239,6025,5867,5764,5599,5472,5353,5249,5087,5022,6053,5843,5715,5606,5445,5363,5250,5137,4971,4937,5886,5705,5590,5463,5323,5253,5161,5028,4891,4859,5759,5584,5453,5374,5260,5148,5034,4967,4811,4735,5611,5434,5342,5271,5158,5041,4978,4863,4716,4646,5476,5326,5248,5153,5056,4974,4871,4766,4615,4557,6737,6548,6365,6171,5919,5778,5648,5487,5403,5345,6544,6368,6183,6009,5788,5666,5517,5370,5294,5220,6383,6209,6020,5859,5688,5557,5411,5308,5201,5110,6217,6020,5892,5755,5566,5433,5326,5208,5107,5016,6044,5865,5757,5600,5440,5350,5240,5110,5020,4957,5889,5721,5593,5474,5345,5221,5116,5010,4930,4828,5780,5562,5463,5378,5224,5121,5033,4926,4841,4706,5638,5447,5366,5267,5134,5034,4978,4835,4744,4643,5509,5343,5257,5139,5067,4964,4833,4742,4644,4535,5400,5216,5132,5067,4972,4881,4777,4648,4568,4450,6548,6351,6164,6006,5754,5634,5465,5426,5320,5239,6361,6186,5983,5869,5643,5484,5387,5335,5192,5121,6191,6016,5887,5717,5522,5402,5290,5219,5093,5044,6045,5898,5721,5591,5436,5327,5179,5139,5022,4964,5902,5750,5601,5496,5331,5210,5098,5060,4932,4854,5737,5578,5488,5381,5195,5120,5006,4956,4835,4752,5613,5461,5364,5243,5110,5022,4914,4847,4713,4643,5521,5349,5239,5159,5043,4949,4833,4773,4629,4541,5404,5228,5146,5067,4930,4843,4756,4659,4531,4446,5268,5144,5055,4975,4874,4757,4645,4569,4442,4368,6333,6158,6011,5846,5592,5458,5428,5300,5215,5144,6166,6010,5860,5705,5491,5373,5322,5200,5128,5052,6037,5859,5706,5599,5395,5251,5201,5138,5024,4937,5880,5721,5609,5484,5287,5163,5135,5020,4932,4874,5758,5613,5503,5364,5207,5083,5035,4955,4841,4793,5621,5489,5366,5249,5123,4990,4914,4867,4763,4654,5500,5330,5246,5162,5015,4900,4842,4742,4643,4562,5403,5247,5155,5044,4932,4837,4744,4639,4540,4464,5291,5162,5060,4942,4860,4750,4651,4548,4459,4371,5206,5037,4966,4875,4752,4654,4580,4467,4365,4286,6139,5975,5814,5684,5457,5404,5305,5224,5148,5038,5988,5827,5679,5556,5330,5299,5210,5138,5045,4964,5845,5711,5576,5437,5232,5220,5112,5002,4969,4894,5728,5581,5451,5341,5170,5117,5014,4949,4859,4784,5622,5470,5355,5259,5069,5024,4939,4875,4778,4694,5464,5351,5249,5148,4967,4932,4855,4760,4650,4589,5364,5246,5151,5022,4906,4847,4714,4660,4579,4477,5293,5137,5026,4967,4824,4744,4654,4560,4481,4397,5190,5048,4960,4859,4719,4652,4526,4472,4353,4274,5081,4970,4874,4758,4628,4593,4454,4356,4306,4207,5938,5797,5661,5522,5374,5310,5230,5133,5048,4979,5799,5678,5535,5397,5294,5197,5130,5038,4966,4891,5704,5549,5426,5297,5192,5091,5014,4995,4880,4781,5565,5418,5321,5214,5075,5007,4952,4849,4777,4693,5472,5346,5213,5107,5032,4929,4866,4764,4675,4598,5381,5223,5105,5005,4939,4832,4751,4677,4600,4501,5276,5114,5023,4943,4811,4718,4662,4566,4471,4414,5163,5057,4949,4822,4737,4624,4548,4481,4401,4325,5085,4942,4860,4727,4644,4537,4456,4365,4294,4226,4999,4872,4757,4647,4567,4429,4387,4304,4190,4129};int Solve(int x) {int ans=0,i;for(i=0;i<=10000;i++){if(x>=i*100000)ans+=r[i];else break;}for(int j=(i-1)*100000+1;j<=x;j++)if(j%f(j)==0) ans++;return ans; }int main() {scanf("%d",&t);for(int ca=1;ca<=t;ca++){scanf("%d%d",&a,&b);printf("Case %d: %d\n",ca,Solve(b)-Solve(a-1));}return 0; }

然后就是數位DP求解：

分別計算出[1, b]中符合條件的個數和[1, a-1]中符合條件的個數。

d[l][i][j][k]表示前l(fā)位和為i模j的結果為k的數的個數，那么就有方程

d[l+1][i+x][j][(k*10+x)%j] += d[l][i][j][k]，預處理出d[l][i][j][k]，然后再逐位統計即可。

#include <iostream> #include <string.h> #include <stdio.h>using namespace std;int dp[10][82][82][82];void Pre() {int i,j,k,l,x;for(i=1;i<=81;i++) dp[0][0][i][0]=1;for(l=0; l<9; l++)for(i=0; i<=l*9; i++)for(j=1; j<=81; j++)for(k=0; k<j; k++)for(x=0; x<=9; x++){dp[l+1][i+x][j][(k*10+x)%j]+=dp[l][i][j][k];} }int Solve(int x) {if(x==0) return 0;int ret=0;if(x==1000000000){ret++;x--;}int i,j,k;int len,sum,t,tt,power=1,bit[11];len=sum=0;t=tt=x;while(t){bit[++len]=t%10;t/=10;sum+=bit[len];}if(x%sum==0) ret++;for(i=1;i<=len;i++){tt/=10;power*=10;t=tt*power;sum-=bit[i];for(j=0;j<bit[i];j++){for(k=sum+j;k<=sum+j+9*(i-1);k++){if(k==0) continue;int dd=t%k;if(dd>0) dd=k-dd;ret+=dp[i-1][k-sum-j][k][dd];}t+=power/10;}}return ret; }int main() {int ta=1,cas;Pre();scanf("%d",&cas);while(cas--){int a,b;scanf("%d%d",&a,&b);printf("Case %d: %d\n",ta++,Solve(b)-Solve(a-1));}return 0; }

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總結

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