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# Mods!

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132
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How many moduli between 1 and 1,000 that have zero remainder does the number 25! have? Just to be clear:
25! mod 1, 2, 3, 4......1,000 =0. How many of them are there between 1 and 1,000? Thank you for help.

Oct 22, 2018

#1
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I don't know if there is a shortcut or some formula to calculate this? Maybe somebody else on this forum does.

However, it is very easy to write a very short and simple computer code to check all the numbers from 1 up to 1,000. When you do that, this is what you get:

25! mod [1 to 1,000] =0 are the following numbers:

(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, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 91, 92, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 119, 120, 121, 125, 126, 128, 130, 132, 133, 135, 136, 138, 140, 143, 144, 147, 150, 152, 153, 154, 156, 160, 161, 162, 165, 168, 170, 171, 175, 176, 180, 182, 184, 187, 189, 190, 192, 195, 196, 198, 200, 204, 207, 208, 209, 210, 216, 220, 221, 224, 225, 228, 230, 231, 234, 238, 240, 242, 243, 245, 247, 250, 252, 253, 255, 256, 260, 264, 266, 270, 272, 273, 275, 276, 280, 285, 286, 288, 294, 297, 299, 300, 304, 306, 308, 312, 315, 320, 322, 323, 324, 325, 330, 336, 340, 342, 343, 345, 350, 351, 352, 357, 360, 363, 364, 368, 374, 375, 378, 380, 384, 385, 390, 391, 392, 396, 399, 400, 405, 408, 414, 416, 418, 420, 425, 429, 432, 437, 440, 441, 442, 448, 450, 455, 456, 459, 460, 462, 468, 475, 476, 480, 483, 484, 486, 490, 494, 495, 500, 504, 506, 510, 512, 513, 520, 525, 528, 532, 539, 540, 544, 546, 550, 552, 560, 561, 567, 570, 572, 575, 576, 585, 588, 594, 595, 598, 600, 605, 608, 612, 616, 621, 624, 625, 627, 630, 637, 640, 644, 646, 648, 650, 660, 663, 665, 672, 675, 680, 684, 686, 690, 693, 700, 702, 704, 714, 715, 720, 726, 728, 729, 735, 736, 741, 748, 750, 756, 759, 760, 765, 768, 770, 780, 782, 784, 792, 798, 800, 805, 810, 816, 819, 825, 828, 832, 833, 836, 840, 847, 850, 855, 858, 864, 874, 875, 880, 882, 884, 891, 896, 897, 900, 910, 912, 918, 920, 924, 931, 935, 936, 945, 950, 952, 960, 966, 968, 969, 972, 975, 980, 988, 990, 1000)
Total = 358

Man! That is a lot of numbers!!! If there isn't a formula, how are you supposed to calculate all of this?

Oct 22, 2018