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please help me:

why 1000! answer last digits are zero? help me

 

The factorial of 1000 is calculated as below:


n = 1000

1000! = 1000 × 999 × 998 × 997 × 996 × 995 × 994 × 993 × 992 × 991 × 990 × 989 × 988 × 987 × 986 × 985 × 984 × 983 × 982 × 981 × 980 × 979 × 978 × 977 × 976 × 975 × 974 × 973 × 972 × 971 × 970 × 969 × 968 × 967 × 966 × 965 × 964 × 963 × 962 × 961 × 960 × 959 × 958 × 957 × 956 × 955 × 954 × 953 × 952 × 951 × 950 × 949 × 948 × 947 × 946 × 945 × 944 × 943 × 942 × 941 × 940 × 939 × 938 × 937 × 936 × 935 × 934 × 933 × 932 × 931 × 930 × 929 × 928 × 927 × 926 × 925 × 924 × 923 × 922 × 921 × 920 × 919 × 918 × 917 × 916 × 915 × 914 × 913 × 912 × 911 × 910 × 909 × 908 × 907 × 906 × 905 × 904 × 903 × 902 × 901 × 900 × 899 × 898 × 897 × 896 × 895 × 894 × 893 × 892 × 891 × 890 × 889 × 888 × 887 × 886 × 885 × 884 × 883 × 882 × 881 × 880 × 879 × 878 × 877 × 876 × 875 × 874 × 873 × 872 × 871 × 870 × 869 × 868 × 867 × 866 × 865 × 864 × 863 × 862 × 861 × 860 × 859 × 858 × 857 × 856 × 855 × 854 × 853 × 852 × 851 × 850 × 849 × 848 × 847 × 846 × 845 × 844 × 843 × 842 × 841 × 840 × 839 × 838 × 837 × 836 × 835 × 834 × 833 × 832 × 831 × 830 × 829 × 828 × 827 × 826 × 825 × 824 × 823 × 822 × 821 × 820 × 819 × 818 × 817 × 816 × 815 × 814 × 813 × 812 × 811 × 810 × 809 × 808 × 807 × 806 × 805 × 804 × 803 × 802 × 801 × 800 × 799 × 798 × 797 × 796 × 795 × 794 × 793 × 792 × 791 × 790 × 789 × 788 × 787 × 786 × 785 × 784 × 783 × 782 × 781 × 780 × 779 × 778 × 777 × 776 × 775 × 774 × 773 × 772 × 771 × 770 × 769 × 768 × 767 × 766 × 765 × 764 × 763 × 762 × 761 × 760 × 759 × 758 × 757 × 756 × 755 × 754 × 753 × 752 × 751 × 750 × 749 × 748 × 747 × 746 × 745 × 744 × 743 × 742 × 741 × 740 × 739 × 738 × 737 × 736 × 735 × 734 × 733 × 732 × 731 × 730 × 729 × 728 × 727 × 726 × 725 × 724 × 723 × 722 × 721 × 720 × 719 × 718 × 717 × 716 × 715 × 714 × 713 × 712 × 711 × 710 × 709 × 708 × 707 × 706 × 705 × 704 × 703 × 702 × 701 × 700 × 699 × 698 × 697 × 696 × 695 × 694 × 693 × 692 × 691 × 690 × 689 × 688 × 687 × 686 × 685 × 684 × 683 × 682 × 681 × 680 × 679 × 678 × 677 × 676 × 675 × 674 × 673 × 672 × 671 × 670 × 669 × 668 × 667 × 666 × 665 × 664 × 663 × 662 × 661 × 660 × 659 × 658 × 657 × 656 × 655 × 654 × 653 × 652 × 651 × 650 × 649 × 648 × 647 × 646 × 645 × 644 × 643 × 642 × 641 × 640 × 639 × 638 × 637 × 636 × 635 × 634 × 633 × 632 × 631 × 630 × 629 × 628 × 627 × 626 × 625 × 624 × 623 × 622 × 621 × 620 × 619 × 618 × 617 × 616 × 615 × 614 × 613 × 612 × 611 × 610 × 609 × 608 × 607 × 606 × 605 × 604 × 603 × 602 × 601 × 600 × 599 × 598 × 597 × 596 × 595 × 594 × 593 × 592 × 591 × 590 × 589 × 588 × 587 × 586 × 585 × 584 × 583 × 582 × 581 × 580 × 579 × 578 × 577 × 576 × 575 × 574 × 573 × 572 × 571 × 570 × 569 × 568 × 567 × 566 × 565 × 564 × 563 × 562 × 561 × 560 × 559 × 558 × 557 × 556 × 555 × 554 × 553 × 552 × 551 × 550 × 549 × 548 × 547 × 546 × 545 × 544 × 543 × 542 × 541 × 540 × 539 × 538 × 537 × 536 × 535 × 534 × 533 × 532 × 531 × 530 × 529 × 528 × 527 × 526 × 525 × 524 × 523 × 522 × 521 × 520 × 519 × 518 × 517 × 516 × 515 × 514 × 513 × 512 × 511 × 510 × 509 × 508 × 507 × 506 × 505 × 504 × 503 × 502 × 501 × 500 × 499 × 498 × 497 × 496 × 495 × 494 × 493 × 492 × 491 × 490 × 489 × 488 × 487 × 486 × 485 × 484 × 483 × 482 × 481 × 480 × 479 × 478 × 477 × 476 × 475 × 474 × 473 × 472 × 471 × 470 × 469 × 468 × 467 × 466 × 465 × 464 × 463 × 462 × 461 × 460 × 459 × 458 × 457 × 456 × 455 × 454 × 453 × 452 × 451 × 450 × 449 × 448 × 447 × 446 × 445 × 444 × 443 × 442 × 441 × 440 × 439 × 438 × 437 × 436 × 435 × 434 × 433 × 432 × 431 × 430 × 429 × 428 × 427 × 426 × 425 × 424 × 423 × 422 × 421 × 420 × 419 × 418 × 417 × 416 × 415 × 414 × 413 × 412 × 411 × 410 × 409 × 408 × 407 × 406 × 405 × 404 × 403 × 402 × 401 × 400 × 399 × 398 × 397 × 396 × 395 × 394 × 393 × 392 × 391 × 390 × 389 × 388 × 387 × 386 × 385 × 384 × 383 × 382 × 381 × 380 × 379 × 378 × 377 × 376 × 375 × 374 × 373 × 372 × 371 × 370 × 369 × 368 × 367 × 366 × 365 × 364 × 363 × 362 × 361 × 360 × 359 × 358 × 357 × 356 × 355 × 354 × 353 × 352 × 351 × 350 × 349 × 348 × 347 × 346 × 345 × 344 × 343 × 342 × 341 × 340 × 339 × 338 × 337 × 336 × 335 × 334 × 333 × 332 × 331 × 330 × 329 × 328 × 327 × 326 × 325 × 324 × 323 × 322 × 321 × 320 × 319 × 318 × 317 × 316 × 315 × 314 × 313 × 312 × 311 × 310 × 309 × 308 × 307 × 306 × 305 × 304 × 303 × 302 × 301 × 300 × 299 × 298 × 297 × 296 × 295 × 294 × 293 × 292 × 291 × 290 × 289 × 288 × 287 × 286 × 285 × 284 × 283 × 282 × 281 × 280 × 279 × 278 × 277 × 276 × 275 × 274 × 273 × 272 × 271 × 270 × 269 × 268 × 267 × 266 × 265 × 264 × 263 × 262 × 261 × 260 × 259 × 258 × 257 × 256 × 255 × 254 × 253 × 252 × 251 × 250 × 249 × 248 × 247 × 246 × 245 × 244 × 243 × 242 × 241 × 240 × 239 × 238 × 237 × 236 × 235 × 234 × 233 × 232 × 231 × 230 × 229 × 228 × 227 × 226 × 225 × 224 × 223 × 222 × 221 × 220 × 219 × 218 × 217 × 216 × 215 × 214 × 213 × 212 × 211 × 210 × 209 × 208 × 207 × 206 × 205 × 204 × 203 × 202 × 201 × 200 × 199 × 198 × 197 × 196 × 195 × 194 × 193 × 192 × 191 × 190 × 189 × 188 × 187 × 186 × 185 × 184 × 183 × 182 × 181 × 180 × 179 × 178 × 177 × 176 × 175 × 174 × 173 × 172 × 171 × 170 × 169 × 168 × 167 × 166 × 165 × 164 × 163 × 162 × 161 × 160 × 159 × 158 × 157 × 156 × 155 × 154 × 153 × 152 × 151 × 150 × 149 × 148 × 147 × 146 × 145 × 144 × 143 × 142 × 141 × 140 × 139 × 138 × 137 × 136 × 135 × 134 × 133 × 132 × 131 × 130 × 129 × 128 × 127 × 126 × 125 × 124 × 123 × 122 × 121 × 120 × 119 × 118 × 117 × 116 × 115 × 114 × 113 × 112 × 111 × 110 × 109 × 108 × 107 × 106 × 105 × 104 × 103 × 102 × 101 × 100 × 99 × 98 × 97 × 96 × 95 × 94 × 93 × 92 × 91 × 90 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81 × 80 × 79 × 78 × 77 × 76 × 75 × 74 × 73 × 72 × 71 × 70 × 69 × 68 × 67 × 66 × 65 × 64 × 63 × 62 × 61 × 60 × 59 × 58 × 57 × 56 × 55 × 54 × 53 × 52 × 51 × 50 × 49 × 48 × 47 × 46 × 45 × 44 × 43 × 42 × 41 × 40 × 39 × 38 × 37 × 36 × 35 × 34 × 33 × 32 × 31 × 30 × 29 × 28 × 27 × 26 × 25 × 24 × 23 × 22 × 21 × 20 × 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

1000! = 402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

 Feb 24, 2023
 #1
avatar
+1

Because when factor it, you get this:

 

1000! = 2^994 * 3^498 * 5^249 * 7^164 * 11^98 * 13^81 * 17^61 * 19^54 * 23^44 * 29^35 * 31^33 * 37^27 * 41^24 * 43^23 * 47^21 * 53^18 * 59^16 * 61^16 * 67^14 * 71^14 * 73^13 * 79^12 * 83^12 * 89^11 * 97^10 * 101^9 * 103^9 * 107^9 * 109^9 * 113^8 * 127^7 * 131^7 * 137^7 * 139^7 * 149^6 * 151^6 * 157^6 * 163^6 * 167^5 * 173^5 * 179^5 * 181^5 * 191^5 * 193^5 * 197^5 * 199^5 * 211^4 * 223^4 * 227^4 * 229^4 * 233^4 * 239^4 * 241^4 * 251^3 * 257^3 * 263^3 * 269^3 * 271^3 * 277^3 * 281^3 * 283^3 * 293^3 * 307^3 * 311^3 * 313^3 * 317^3 * 331^3 * 337^2 * 347^2 * 349^2 * 353^2 * 359^2 * 367^2 * 373^2 * 379^2 * 383^2 * 389^2 * 397^2 * 401^2 * 409^2 * 419^2 * 421^2 * 431^2 * 433^2 * 439^2 * 443^2 * 449^2 * 457^2 * 461^2 * 463^2 * 467^2 * 479^2 * 487^2 * 491^2 * 499^2 * 503 * 509 * 521 * 523 * 541 * 547 * 557 * 563 * 569 * 571 * 577 * 587 * 593 * 599 * 601 * 607 * 613 * 617 * 619 * 631 * 641 * 643 * 647 * 653 * 659 * 661 * 673 * 677 * 683 * 691 * 701 * 709 * 719 * 727 * 733 * 739 * 743 * 751 * 757 * 761 * 769 * 773 * 787 * 797 * 809 * 811 * 821 * 823 * 827 * 829 * 839 * 853 * 857 * 859 * 863 * 877 * 881 * 883 * 887 * 907 * 911 * 919 * 929 * 937 * 941 * 947 * 953 * 967 * 971 * 977 * 983 * 991 * 997

 

You see 5^249? That is 249 "5s" that are multiplied by 249 "2s". And: 5 x 2 =10 and you get 249 zeros at the end! And that is why.

 Feb 24, 2023
 #2
avatar
0

perracita, you could have just said it is divisible by 10 goddamit

Guest Feb 24, 2023

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