A natural number less than 1000 has exactly 9 factors. How many such numbers exist?
Answers
Answer:
Step-by-step explanation:
I counted 257 numbers less than 1000 which only have 3 factors. I did not include 1 as a factor as that seemed almost trivial to figure out. Each factor is a prime, so a number’s prime factorization would only have 3 numbers to fit the criterion. I’ve searched for the combinatorial solution but only for a little bit of time. We have 3 slots, starting with slot 1 we can have 2 3 5 or 7
2–2-any prime between 2 and 241 53 primes 8 12 20 28 44 52 68 76 92 116 124 148 164 172 188 212 236 244 268 284 292 316 332 356 388 404 412 428 436 452 508 524 548 556 596 604 628 652 668 692 716 724 764 772 788 796 844 892 908 916 932 956 964
2–3-any prime between 3 and 163 37 primes 18 30 42 66 78 102 114 138 174 186 222 246 258 282 318 354 366 402 426 438 474 498 534 582 606 618 642 654 678 762 786 822 834 894 906 942 978
2–5-any prime between 5 and 97 23 primes 50 70 110 130 170 190 230 290 310 370 410 430 470 530 590 610 670 710 730 790 830 890 970
2–7-any prime between 7 and 71 17 primes 98 154 182 238 266 322 406 434 518 574 602 658 742 826 854 938 994
2–11-any prime between 11 and 43 10 primes 242 286 374 418 506 638 682 814 902 946
2–13-any prime between 13 and 37 7 primes 338 442 494 598 754 806 962
2–17-any prime between 17 and 29 4 primes 578 646 714 986
2–19–19 or 23 2 primes 722 798
3–3-any prime between 3 and 109 28 primes 27 45 63 99 117 153 171 207 261 279 333 369 387 423 477 531 549 603 639 657 711 747 801 873 909 927 963 981
3–5-any prime between 5 and 61 16 primes 75 105 165 195 255 285 345 435 465 555 615 645 705 795 885 915
3–7-any prime between 7 and 47 12 primes 147 231 273 357 399 483 609 651 777 861 903 987
3–11-any prime between 11 and 29 6 primes 363 429 561 627 759 957
3–13-any prime between 13 and 23 4 primes 507 663 741 897
3–17–17 or 19 2 primes 807 969
5–5-any prime between 5 and 37 10 primes 125 175 275 325 425 475 575 725 775 925
5–7-any prime between 7 and 23 6 primes 245 385 455 595 665 805
5–11-any prime between 11 and 17 3 primes 605 715 935
5–13–13 1 prime 845
7–7-any prime between 7 and 19 5 primes 343 539 637 833 931
7–11–11 1 prime 847