ModEquation

SRM 704 · 2016-12-04 · by cgy4ever

SRM 704 · 2016-12-04 · by cgy4ever · Dynamic Programming, Math

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Class Name: ModEquation

Return Type: int[]

Method Name: count

Arg Types: (int, int, vector<int>)

Problem Statement

This problem has a non-standard time limit: 3 seconds.

You are given the ints n and K.

Consider the following modular equation with n variables: (x[0] * x[1] * x[2] * ... * x[n-1]) mod K = v. How many solutions does it have for a given value v?

Formally, we want to find the number of sequences (x[0], x[1], ..., x[n-1]) such that each x[i] is an integer between 0 and K-1, inclusive, and the product of all x[i] gives the remainder v when divided by K.

You are also given a int[] query containing q elements. For each i, let answer[i] be the number of solutions the above equation has when v = query[i].

Return a int[] with q elements. For each i, element i of the return value should be the number (answer[i] mod 10^9 + 7).

Constraints

n will be between 1 and 50, inclusive.
K will be between 2 and 1,000,000,000, inclusive.
query will contain between 1 and 1,000 elements, inclusive.
Each element in query will be between 0 and (K-1), inclusive.

Examples

2
2
{0,1}
Returns: {3, 1 }

The correct return value should have two elements: Element 0 of the return value should be the number of solutions of the equation (x[0] * x[1]) mod 2 = 0. Element 1 of the return value should be the number of solutions of the equation (x[0] * x[1]) mod 2 = 1. We have: (0 * 0) % 2 = 0 (0 * 1) % 2 = 0 (1 * 0) % 2 = 0 (1 * 1) % 2 = 1 So, the first equation has 3 solutions and the second equation has 1 solution.

2
4
{0,1,2,3}
Returns: {8, 2, 4, 2 }

6
6
{4}
Returns: {2016 }

1
2
{0,0,0,1,1,1}
Returns: {1, 1, 1, 1, 1, 1 }

50
931170240
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 888, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998, 999}
Returns: {185422162, 284971463, 248573052, 248573052, 338612784, 489795299, 428652516, 990099617, 869288221, 418560186, 489764782, 726753547, 930639088, 391324527, 504980507, 489764782, 768068853, 857367656, 928009160, 145007081, 489001857, 504980507, 337677098, 284971463, 464410749, 489795299, 566226217, 418560186, 377002841, 284971463, 488238932, 284971463, 95143555, 337677098, 868382506, 807541628, 664233419, 284971463, 250354001, 566226217, 476032132, 284971463, 249025175, 284971463, 610765943, 585955946, 248573052, 284971463, 403442384, 990099617, 489764782, 868382506, 938768439, 284971463, 928009160, 603617600, 201382533, 250354001, 248573052, 284971463, 450092682, 284971463, 248573052, 561622474, 318375630, 556035130, 883854788, 284971463, 143753749, 248573052, 377081120, 284971463, 180045850, 284971463, 248573052, 489764782, 884026987, 260898601, 311310654, 284971463, 307425707, 418560186, 248573052, 284971463, 850141924, 438726773, 248573052, 248573052, 253276273, 284971463, 297797097, 796163001, 338612784, 248573052, 248573052, 190750119, 757177722, 284971463, 504980507, 403746906, 489001857, 284971463, 419124999, 284971463, 271986164, 377081120, 248573052, 284971463, 664233419, 284971463, 180879790, 248573052, 918318550, 284971463, 517699966, 489795299, 338612784, 473195155, 248573052, 358526761, 801606439, 726753547, 248573052, 248573052, 338612784, 489795299, 81123504, 284971463, 318375630, 248573052, 801756311, 284971463, 538296940, 474303307, 248573052, 585955946, 825064971, 284971463, 428652516, 284971463, 615568497, 248573052, 248573052, 149142619, 885607502, 489795299, 248573052, 504980507, 338612784, 284971463, 488238932, 284971463, 656467672, 778654309, 44929959, 489795299, 938421628, 284971463, 248573052, 248573052, 520197616, 990099617, 928009160, 284971463, 338612784, 180879790, 248573052, 284971463, 69126580, 391324527, 936338503, 584305975, 338612784, 284971463, 428652516, 807541628, 605910598, 248573052, 248573052, 284971463, 93825921, 284971463, 808149777, 248573052, 869288221, 489795299, 428652516, 880811527, 338612784, 561622474, 537505887, 284971463, 918781395, 284971463, 248573052, 801756311, 377002841, 284971463, 187345160, 284971463, 476032132, 248573052, 248573052, 990099617, 187687401, 489795299, 248573052, 418560186, 603816652, 911833651, 854055874, 284971463, 338612784, 248573052, 248573052, 489795299, 180045850, 990099617, 248573052, 248573052, 612434617, 835714237, 428652516, 284971463, 917840277, 585955946, 248573052, 284971463, 201349042, 284971463, 489764782, 44929959, 869288221, 284971463, 659757589, 489795299, 338612784, 248573052, 926337931, 284971463, 371285245, 284971463, 337677098, 418560186, 338612784, 807541628, 428652516, 454273121, 869288221, 248573052, 489764782, 284971463, 68649338, 726753547, 248573052, 936338503, 318375630, 284971463, 428652516, 990099617, 944785794, 418560186, 248573052, 284971463, 663813566, 489795299, 715165189, 248573052, 338612784, 284971463, 297797097, 284971463, 682110794, 808149777, 248573052, 603617600, 930639088, 284971463, 248573052, 418560186, 669853878, 284971463, 428652516, 284971463, 338612784, 537505887, 457130901, 990099617, 764560960, 857367656, 489764782, 248573052, 338612784, 284971463, 249025175, 489795299, 869288221, 403746906, 248573052, 391324527, 450092682, 990099617, 248573052, 248573052, 698196598, 489795299, 932715184, 284971463, 645713951, 248573052, 489764782, 284971463, 599308109, 284971463, 248573052, 496327138, 338612784, 284971463, 428652516, 726753547, 921001829, 248573052, 504980507, 938888706, 664233419, 556035130, 248573052, 248573052, 869288221, 990099617, 43989437, 284971463, 338612784, 418560186, 248573052, 489795299, 915927185, 284971463, 566226217, 248573052, 376631662, 726753547, 215298547, 990099617, 869288221, 489764782, 248573052, 284971463, 930639088, 284971463, 377081120, 473195155, 343834415, 284971463, 428652516, 489795299, 338612784, 926337931, 248573052, 284971463, 626315957, 145007081, 248573052, 337677098, 107819170, 489795299, 428652516, 284971463, 768068853, 418560186, 489764782, 990099617, 930639088, 284971463, 40576042, 489764782, 869288221, 391324527, 81123504, 284971463, 206400024, 248573052, 248573052, 284971463, 918781395, 391713307, 248573052, 418560186, 338612784, 284971463, 87815270, 857367656, 201382533, 248573052, 248573052, 489795299, 777301552, 284971463, 248573052, 715165189, 307425707, 284971463, 428652516, 391324527, 338612784, 585955946, 504980507, 726753547, 253248263, 284971463, 489764782, 248573052, 338612784, 990099617, 928009160, 489795299, 721219801, 248573052, 591682235, 284971463, 778424640, 284971463, 248573052, 418560186, 869288221, 438726773, 428652516, 990099617, 338612784, 457130901, 489764782, 284971463, 885607502, 284971463, 504980507, 489764782, 338612784, 145007081, 428652516, 284971463, 948866627, 561622474, 785711563, 284971463, 930639088, 489795299, 248573052, 248573052, 956498525, 284971463, 297797097, 726753547, 338612784, 248573052, 248573052, 703078623, 823383376, 284971463, 248573052, 778654309, 489001857, 284971463, 246497936, 284971463, 768068853, 489764782, 248573052, 284971463, 323818404, 990099617, 489764782, 248573052, 869288221, 726753547, 428652516, 190750119, 121617076, 418560186, 248573052, 284971463, 9880618, 391324527, 248573052, 504980507, 610765943, 489795299, 928009160, 284971463, 869288221, 248573052, 377081120, 284971463, 930639088, 857367656, 713655896, 903812130, 768068853, 990099617, 428652516, 284971463, 489001857, 248573052, 248573052, 284971463, 856588523, 489795299, 337677098, 566226217, 338612784, 284971463, 816924828, 990099617, 318375630, 584305975, 248573052, 489795299, 930639088, 726753547, 504980507, 248573052, 376286984, 284971463, 928009160, 284971463, 338612784, 377081120, 248573052, 857367656, 295529690, 284971463, 489764782, 418560186, 736712197, 391324527, 428652516, 489795299, 869288221, 248573052, 248573052, 260898601, 93825921, 284971463, 248573052, 248573052, 766796529, 489795299, 407488570, 284971463, 338612784, 418560186, 180879790, 145007081, 464410749, 990099617, 248573052, 489764782, 338612784, 284971463, 928009160, 391324527, 375563824, 40576042, 248573052, 284971463, 930639088, 489795299, 248573052, 561622474, 869288221, 284971463, 875294168, 284971463, 156837895, 248573052, 504980507, 489795299, 29113224, 284971463, 868382506, 248573052, 489001857, 990099617, 428652516, 726753547, 869288221, 290590256, 248573052, 284971463, 850141924, 145007081, 489764782, 248573052, 768068853, 284971463, 187345160, 808148146, 338612784, 248573052, 566226217, 284971463, 801606439, 284971463, 504980507, 418560186, 338612784, 603617600, 428652516, 284971463, 340523208, 504980507, 489764782, 391324527, 784237031, 284971463, 248573052, 489764782, 859041745, 284971463, 428652516, 284971463, 489001857, 418560186, 248573052, 990099617, 190832390, 489795299, 248573052, 591682235, 338612784, 857367656, 816356732, 284971463, 869288221, 248573052, 248573052, 489795299, 930639088, 796163001, 337677098, 418560186, 921001829, 284971463, 428652516, 284971463, 377002841, 489764782, 944434978, 284971463, 180045850, 726753547, 801756311, 504980507, 338612784, 284971463, 428652516, 489795299, 768068853, 418560186, 504980507, 284971463, 621730640, 284971463, 248573052, 785711563, 869288221, 175395716, 928009160, 284971463, 338612784, 248573052, 489764782, 726753547, 892013535, 284971463, 248573052, 585955946, 938768439, 284971463, 428652516, 990099617, 861615435, 248573052, 337677098, 284971463, 990112917, 489795299, 504980507, 248573052, 768068853, 391324527, 488238932, 284971463, 338612784, 590524748, 248573052, 489795299, 464410749, 857367656, 248573052, 248573052, 615568497, 284971463, 659757589, 145007081, 984652543, 489764782, 248573052, 990099617, 930639088, 284971463, 489764782, 418560186, 869288221, 284971463, 316896228, 231755871, 338612784, 248573052, 248573052, 284971463, 548686376, 990099617, 250354001, 248573052, 338612784, 489795299, 883854788, 284971463, 202198935, 418560186, 489764782, 857367656, 930639088, 284971463, 248573052, 377081120, 95143555, 726753547, 928009160, 284971463, 489001857, 713655896, 504980507, 284971463, 464410749, 489795299, 248573052, 418560186, 34689064, 990099617, 488238932, 284971463, 768068853, 248573052, 566226217, 489795299, 68649338, 284971463, 248573052, 337677098, 577600395, 284971463, 428652516, 990099617, 338612784, 871483055, 248573052, 391324527, 918781395, 284971463, 585665217, 248573052, 338612784, 284971463, 928009160, 489795299, 869288221, 504980507, 248573052, 145007081, 239289371, 726753547, 868382506, 418560186, 918318550, 489795299, 428652516, 284971463, 338612784, 248573052, 489764782, 990099617, 806560146, 391324527, 248573052, 489764782, 338612784, 284971463, 758259205, 857367656, 520197616, 418560186, 248573052, 726753547, 930639088, 807541628, 566226217, 248573052, 869288221, 284971463, 297797097, 284971463, 377002841, 248573052, 337677098, 489795299, 105539462, 145007081, 248573052, 414076701, 489001857, 284971463, 428652516, 284971463, 869288221, 180879790, 504980507, 284971463, 664233419, 284971463, 489764782, 248573052, 726766804, 358526761, 428652516, 489795299, 587896891, 418560186, 248573052, 284971463, 492693669, 284971463, 248573052, 248573052, 338612784, 556035130, 928009160, 260898601, 768068853, 248573052, 936338503, 284971463, 930639088, 284971463, 504980507, 697458893, 869288221, 284971463, 856544896, 284971463, 489001857, 504980507, 248573052, 284971463, 764560960, 489795299, 248573052, 868382506, 377002841, 726753547, 488238932, 391324527, 869288221, 418560186, 250354001, 807541628, 930639088, 284971463, 248573052, 248573052, 322482722, 284971463, 81123504, 284971463, 535644720, 489764782, 248573052, 284971463, 464410749, 990099617, 489764782, 403746906, 338612784, 145007081, 428652516, 489795299, 956498525, 566226217, 248573052, 284971463, 93825921, 857367656, 337677098, 504980507, 869288221, 489795299, 428652516, 284971463, 338612784, 418560186, 153930905, 284971463, 909829662, 726753547, 248573052, 489764782, 338612784, 990099617, 932715184, 284971463, 476032132, 248573052, 248573052, 391324527, 285697326, 489795299, 248573052, 418560186, 95143555, 284971463, 488238932, 474303307, 338612784, 248573052, 248573052, 488609897, 612852301, 284971463, 504980507, 248573052, 489001857, 284971463, 428652516, 284971463, 768068853, 496327138, 337677098, 284971463, 930639088, 391324527, 537505887, 248573052, 441362639, 284971463, 928009160, 489795299, 338612784, 337677098, 248573052, 990099617, 50091128, 284971463, 566226217, 418560186, 338612784, 489795299, 249025175, 284971463, 253276273, 944434978, 489764782, 284971463, 664233419, 990099617, 248573052, 801756311, 768068853, 284971463, 428652516, 726753547, 615568497, 418560186, 248573052, 284971463, 464410749, 489795299, 868382506, 504980507, 198225222, 284971463, 190606185, 284971463, 95143555, 248573052, 504980507, 489795299, 930639088, 284971463, 248573052, 418560186 }

Submissions are judged against all 109 archived test cases, of which 5 are shown here. Case numbers match the judge’s.

Coding Area

Language: C++17 · define a public class ModEquation with a public method vector<int> count(int n, int K, vector<int> query) · 109 test cases · 2 s / 256 MB per case