# Composite number

A comprehensive video explaining composite numbers: https://youtu.be/SMOGYNYDSBw

A composite number is a positive integer with at least one divisor different from 1 and itself. Some composite numbers are $4=2^2$ and $12=2\times 6=3\times 4$. Composite numbers atleast have 2 distinct prime divisors.

Note that the number one is neither prime nor composite. It follows that two is the only even prime number, three is the only multiple of three that is prime, and so on.

Every positive integer either is prime, composite, or 1.

Extra: A list of composite numbers from 1 to 1000:

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

"It is interesting to note that every positive integer except 1,2,3,4,5,6,7,9,11 can be written as the sum of 2 comopsite numbers"

### Proof:

A positive integer is either odd or even

Case 1: The positive integer is even.

Any even integer greater than 4 can be written as '4+2n'

2n is composite for n greater than 1 and 4 is aldready a composite number

⇒ 4+2(2), 4+2(3), 4+2(4)..... all can be represented as sum of 2 composte numbers

⇒ 8,10,12...... all are sum of 2 composites. Hence, any even positive integer ≥ 8 can be represented as the sum of 2 composite numbers.

The remaining even positive integers are 2,4 and 6 checking for each manually we get none of 2,4 and 6 can be expressed as sum of 2 composite positive integers So, this concludes that all even positive integers except 2,4 and 64 are sum of 2 composite numbers

Case 2: The positive integer is odd

Any odd integer greater than 9 can be written as ' 9+2n'

2n is composite for n greater than 1 and 9 is already a composite number

⇒ 9+2(2), 9+2(3), 9+2(4)..... all can be represented as sum of 2 composite numbers

⇒ 13,15,17...... all are sum of 2 composites Hence, any odd positive ≥ 13 can be represented as the sum of 2 composite numbers.

The remaining odd positive integers are 1,3,5,7,9 and 11 checking for each manually we get none can be expressed as sum of 2 composite positive integers So, this concludes that all odd positive integers except 3,5,7,9 and 11 are sum of 2 composite numbers.

The Above cases conclude that every positive integers except $1,2,3,4,5,6,7,9,11$ can be expressed as sum of 2 composite numbers

~ proof by SANSGANKRSNGUPTA