+0  
 
+1
122
4
avatar+91 

How many 3-digit numbers are there that have exactly 16 divisors?

 Jun 22, 2022
 #1
avatar
+1

120 = (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120) > 16 (divisors)
168 = (1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168) > 16 (divisors)
210 = (1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210) > 16 (divisors)
216 = (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216) > 16 (divisors)
264 = (1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 66, 88, 132, 264) > 16 (divisors)
270 = (1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270) > 16 (divisors)
280 = (1, 2, 4, 5, 7, 8, 10, 14, 20, 28, 35, 40, 56, 70, 140, 280) > 16 (divisors)
312 = (1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, 312) > 16 (divisors)
330 = (1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165, 330) > 16 (divisors)
378 = (1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, 378) > 16 (divisors)
384 = (1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 384) > 16 (divisors)
390 = (1, 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, 195, 390) > 16 (divisors)
408 = (1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408) > 16 (divisors)
440 = (1, 2, 4, 5, 8, 10, 11, 20, 22, 40, 44, 55, 88, 110, 220, 440) > 16 (divisors)
456 = (1, 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, 228, 456) > 16 (divisors)
462 = (1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462) > 16 (divisors)
510 = (1, 2, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, 170, 255, 510) > 16 (divisors)
520 = (1, 2, 4, 5, 8, 10, 13, 20, 26, 40, 52, 65, 104, 130, 260, 520) > 16 (divisors)
546 = (1, 2, 3, 6, 7, 13, 14, 21, 26, 39, 42, 78, 91, 182, 273, 546) > 16 (divisors)
552 = (1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552) > 16 (divisors)
570 = (1, 2, 3, 5, 6, 10, 15, 19, 30, 38, 57, 95, 114, 190, 285, 570) > 16 (divisors)
594 = (1, 2, 3, 6, 9, 11, 18, 22, 27, 33, 54, 66, 99, 198, 297, 594) > 16 (divisors)
616 = (1, 2, 4, 7, 8, 11, 14, 22, 28, 44, 56, 77, 88, 154, 308, 616) > 16 (divisors)
640 = (1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 320, 640) > 16 (divisors)
680 = (1, 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, 340, 680) > 16 (divisors)
690 = (1, 2, 3, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690) > 16 (divisors)
696 = (1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348, 696) > 16 (divisors)
702 = (1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702) > 16 (divisors)
714 = (1, 2, 3, 6, 7, 14, 17, 21, 34, 42, 51, 102, 119, 238, 357, 714) > 16 (divisors)
728 = (1, 2, 4, 7, 8, 13, 14, 26, 28, 52, 56, 91, 104, 182, 364, 728) > 16 (divisors)
744 = (1, 2, 3, 4, 6, 8, 12, 24, 31, 62, 93, 124, 186, 248, 372, 744) > 16 (divisors)
750 = (1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 125, 150, 250, 375, 750) > 16 (divisors)
760 = (1, 2, 4, 5, 8, 10, 19, 20, 38, 40, 76, 95, 152, 190, 380, 760) > 16 (divisors)
770 = (1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770) > 16 (divisors)
798 = (1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 57, 114, 133, 266, 399, 798) > 16 (divisors)
858 = (1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858) > 16 (divisors)
870 = (1, 2, 3, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, 435, 870) > 16 (divisors)
888 = (1, 2, 3, 4, 6, 8, 12, 24, 37, 74, 111, 148, 222, 296, 444, 888) > 16 (divisors)
896 = (1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 128, 224, 448, 896) > 16 (divisors)
910 = (1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910) > 16 (divisors)
918 = (1, 2, 3, 6, 9, 17, 18, 27, 34, 51, 54, 102, 153, 306, 459, 918) > 16 (divisors)
920 = (1, 2, 4, 5, 8, 10, 20, 23, 40, 46, 92, 115, 184, 230, 460, 920) > 16 (divisors)
930 = (1, 2, 3, 5, 6, 10, 15, 30, 31, 62, 93, 155, 186, 310, 465, 930) > 16 (divisors)
945 = (1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945) > 16 (divisors)
952 = (1, 2, 4, 7, 8, 14, 17, 28, 34, 56, 68, 119, 136, 238, 476, 952) > 16 (divisors)
966 = (1, 2, 3, 6, 7, 14, 21, 23, 42, 46, 69, 138, 161, 322, 483, 966) > 16 (divisors)
984 = (1, 2, 3, 4, 6, 8, 12, 24, 41, 82, 123, 164, 246, 328, 492, 984) > 16 (divisors)

Total of all 3-digit integers = 47

 Jun 23, 2022
 #2
avatar+12 
0

Please tell, by what formula you defined it? thank you

NikOL  Jun 23, 2022
 #3
avatar
0

By this computer code in C++:

 

a=100; b=999;c=(1);p=0; n=2; cycle:if((a % n)==0,c=sort(c,n), 0); n++; if(n ", count c,"(divisors)";p=p+ 1;next:a++;c=1;n=2;if(a<=b, goto cycle, discard=0;print;print"Total of all integers =", p

Guest Jun 23, 2022
 #4
avatar+12 
0

thanks for the clarification

 Jun 27, 2022

24 Online Users

avatar
avatar