Factors of 29,830,680. Calculator of Proper, Improper, Prime and Compound Divisors, if Any

All the factors (divisors) of the number 29,830,680. Connection with the prime factorization of the number

To find all the divisors of the number 29,830,680:

  • 1. Decompose the number into prime factors.
  • See how you can find out how many factors (divisors) the number has, without actually calculating them.
  • 2. Multiply the prime factors in all their unique combinations, that yield different results.

1. Carry out the prime factorization of the number 29,830,680:

The prime factorization of a number: finding the prime numbers that multiply together to make that number.


29,830,680 = 23 × 37 × 5 × 11 × 31
29,830,680 is not a prime number but a composite one.


  • Prime number: a natural number that is divisible (divided evenly) only by 1 and itself. A prime number has exactly two factors: 1 and the number itself.
  • Examples of prime numbers: 2 (factors 1, 2), 3 (factors 1, 3), 5 (factors 1, 5), 7 (factors 1, 7), 11 (factors 1, 11), 13 (factors 1, 13), ...
  • Composite number: a natural number that has at least one factor other than 1 and itself. So it is neither a prime number nor 1.
  • Examples of composite numbers: 4 (it has 3 factors: 1, 2, 4), 6 (it has 4 factors: 1, 2, 3, 6), 8 (it has 4 factors: 1, 2, 4, 8), 9 (it has 3 factors: 1, 3, 9), 10 (it has 4 factors: 1, 2, 5, 10), 12 (it has 6 factors: 1, 2, 3, 4, 6, 12), ...

  • » Online calculator. Check whether a number is prime or not. The prime factorization of composite numbers (decomposition into prime factors)


How to count the number of factors of a number?

Without actually finding the factors

  • If a number N is prime factorized as:
    N = am × bk × cz
    where a, b, c are the prime factors and m, k, z are their exponents, natural numbers, ....
  • ...
  • Then the number of factors of the number N can be calculated as:
    n = (m + 1) × (k + 1) × (z + 1)
  • ...
  • In our case, the number of factors is calculated as:
  • n = (3 + 1) × (7 + 1) × (1 + 1) × (1 + 1) × (1 + 1) = 4 × 8 × 2 × 2 × 2 = 256

But to actually calculate the factors, see below...

2. Multiply the prime factors of the number 29,830,680

  • Multiply the prime factors involved in the prime factorization of the number in all their unique combinations, that give different results.
  • Also consider the exponents of these prime factors.
  • Also add 1 to the list of factors (divisors). All the numbers are divisible by 1.

All the factors (divisors) are listed below - in ascending order

The list of factors (divisors):

Numbers other than 1 that are not prime factors are composite factors (divisors).

neither prime nor composite = 1
prime factor = 2
prime factor = 3
composite factor = 22 = 4
prime factor = 5
composite factor = 2 × 3 = 6
composite factor = 23 = 8
composite factor = 32 = 9
composite factor = 2 × 5 = 10
prime factor = 11
composite factor = 22 × 3 = 12
composite factor = 3 × 5 = 15
composite factor = 2 × 32 = 18
composite factor = 22 × 5 = 20
composite factor = 2 × 11 = 22
composite factor = 23 × 3 = 24
composite factor = 33 = 27
composite factor = 2 × 3 × 5 = 30
prime factor = 31
composite factor = 3 × 11 = 33
composite factor = 22 × 32 = 36
composite factor = 23 × 5 = 40
composite factor = 22 × 11 = 44
composite factor = 32 × 5 = 45
composite factor = 2 × 33 = 54
composite factor = 5 × 11 = 55
composite factor = 22 × 3 × 5 = 60
composite factor = 2 × 31 = 62
composite factor = 2 × 3 × 11 = 66
composite factor = 23 × 32 = 72
composite factor = 34 = 81
composite factor = 23 × 11 = 88
composite factor = 2 × 32 × 5 = 90
composite factor = 3 × 31 = 93
composite factor = 32 × 11 = 99
composite factor = 22 × 33 = 108
composite factor = 2 × 5 × 11 = 110
composite factor = 23 × 3 × 5 = 120
composite factor = 22 × 31 = 124
composite factor = 22 × 3 × 11 = 132
composite factor = 33 × 5 = 135
composite factor = 5 × 31 = 155
composite factor = 2 × 34 = 162
composite factor = 3 × 5 × 11 = 165
composite factor = 22 × 32 × 5 = 180
composite factor = 2 × 3 × 31 = 186
composite factor = 2 × 32 × 11 = 198
composite factor = 23 × 33 = 216
composite factor = 22 × 5 × 11 = 220
composite factor = 35 = 243
composite factor = 23 × 31 = 248
composite factor = 23 × 3 × 11 = 264
composite factor = 2 × 33 × 5 = 270
composite factor = 32 × 31 = 279
composite factor = 33 × 11 = 297
composite factor = 2 × 5 × 31 = 310
composite factor = 22 × 34 = 324
composite factor = 2 × 3 × 5 × 11 = 330
composite factor = 11 × 31 = 341
composite factor = 23 × 32 × 5 = 360
composite factor = 22 × 3 × 31 = 372
composite factor = 22 × 32 × 11 = 396
composite factor = 34 × 5 = 405
composite factor = 23 × 5 × 11 = 440
composite factor = 3 × 5 × 31 = 465
composite factor = 2 × 35 = 486
composite factor = 32 × 5 × 11 = 495
composite factor = 22 × 33 × 5 = 540
composite factor = 2 × 32 × 31 = 558
composite factor = 2 × 33 × 11 = 594
composite factor = 22 × 5 × 31 = 620
composite factor = 23 × 34 = 648
composite factor = 22 × 3 × 5 × 11 = 660
composite factor = 2 × 11 × 31 = 682
composite factor = 36 = 729
composite factor = 23 × 3 × 31 = 744
composite factor = 23 × 32 × 11 = 792
composite factor = 2 × 34 × 5 = 810
composite factor = 33 × 31 = 837
composite factor = 34 × 11 = 891
composite factor = 2 × 3 × 5 × 31 = 930
composite factor = 22 × 35 = 972
composite factor = 2 × 32 × 5 × 11 = 990
composite factor = 3 × 11 × 31 = 1,023
composite factor = 23 × 33 × 5 = 1,080
composite factor = 22 × 32 × 31 = 1,116
composite factor = 22 × 33 × 11 = 1,188
composite factor = 35 × 5 = 1,215
composite factor = 23 × 5 × 31 = 1,240
composite factor = 23 × 3 × 5 × 11 = 1,320
composite factor = 22 × 11 × 31 = 1,364
composite factor = 32 × 5 × 31 = 1,395
composite factor = 2 × 36 = 1,458
composite factor = 33 × 5 × 11 = 1,485
composite factor = 22 × 34 × 5 = 1,620
composite factor = 2 × 33 × 31 = 1,674
composite factor = 5 × 11 × 31 = 1,705
composite factor = 2 × 34 × 11 = 1,782
composite factor = 22 × 3 × 5 × 31 = 1,860
composite factor = 23 × 35 = 1,944
composite factor = 22 × 32 × 5 × 11 = 1,980
composite factor = 2 × 3 × 11 × 31 = 2,046
composite factor = 37 = 2,187
composite factor = 23 × 32 × 31 = 2,232
composite factor = 23 × 33 × 11 = 2,376
composite factor = 2 × 35 × 5 = 2,430
composite factor = 34 × 31 = 2,511
composite factor = 35 × 11 = 2,673
composite factor = 23 × 11 × 31 = 2,728
composite factor = 2 × 32 × 5 × 31 = 2,790
composite factor = 22 × 36 = 2,916
composite factor = 2 × 33 × 5 × 11 = 2,970
composite factor = 32 × 11 × 31 = 3,069
composite factor = 23 × 34 × 5 = 3,240
composite factor = 22 × 33 × 31 = 3,348
composite factor = 2 × 5 × 11 × 31 = 3,410
composite factor = 22 × 34 × 11 = 3,564
composite factor = 36 × 5 = 3,645
composite factor = 23 × 3 × 5 × 31 = 3,720
composite factor = 23 × 32 × 5 × 11 = 3,960
composite factor = 22 × 3 × 11 × 31 = 4,092
composite factor = 33 × 5 × 31 = 4,185
composite factor = 2 × 37 = 4,374
composite factor = 34 × 5 × 11 = 4,455
composite factor = 22 × 35 × 5 = 4,860
composite factor = 2 × 34 × 31 = 5,022
composite factor = 3 × 5 × 11 × 31 = 5,115
composite factor = 2 × 35 × 11 = 5,346
This list continues below...

... This list continues from above
composite factor = 22 × 32 × 5 × 31 = 5,580
composite factor = 23 × 36 = 5,832
composite factor = 22 × 33 × 5 × 11 = 5,940
composite factor = 2 × 32 × 11 × 31 = 6,138
composite factor = 23 × 33 × 31 = 6,696
composite factor = 22 × 5 × 11 × 31 = 6,820
composite factor = 23 × 34 × 11 = 7,128
composite factor = 2 × 36 × 5 = 7,290
composite factor = 35 × 31 = 7,533
composite factor = 36 × 11 = 8,019
composite factor = 23 × 3 × 11 × 31 = 8,184
composite factor = 2 × 33 × 5 × 31 = 8,370
composite factor = 22 × 37 = 8,748
composite factor = 2 × 34 × 5 × 11 = 8,910
composite factor = 33 × 11 × 31 = 9,207
composite factor = 23 × 35 × 5 = 9,720
composite factor = 22 × 34 × 31 = 10,044
composite factor = 2 × 3 × 5 × 11 × 31 = 10,230
composite factor = 22 × 35 × 11 = 10,692
composite factor = 37 × 5 = 10,935
composite factor = 23 × 32 × 5 × 31 = 11,160
composite factor = 23 × 33 × 5 × 11 = 11,880
composite factor = 22 × 32 × 11 × 31 = 12,276
composite factor = 34 × 5 × 31 = 12,555
composite factor = 35 × 5 × 11 = 13,365
composite factor = 23 × 5 × 11 × 31 = 13,640
composite factor = 22 × 36 × 5 = 14,580
composite factor = 2 × 35 × 31 = 15,066
composite factor = 32 × 5 × 11 × 31 = 15,345
composite factor = 2 × 36 × 11 = 16,038
composite factor = 22 × 33 × 5 × 31 = 16,740
composite factor = 23 × 37 = 17,496
composite factor = 22 × 34 × 5 × 11 = 17,820
composite factor = 2 × 33 × 11 × 31 = 18,414
composite factor = 23 × 34 × 31 = 20,088
composite factor = 22 × 3 × 5 × 11 × 31 = 20,460
composite factor = 23 × 35 × 11 = 21,384
composite factor = 2 × 37 × 5 = 21,870
composite factor = 36 × 31 = 22,599
composite factor = 37 × 11 = 24,057
composite factor = 23 × 32 × 11 × 31 = 24,552
composite factor = 2 × 34 × 5 × 31 = 25,110
composite factor = 2 × 35 × 5 × 11 = 26,730
composite factor = 34 × 11 × 31 = 27,621
composite factor = 23 × 36 × 5 = 29,160
composite factor = 22 × 35 × 31 = 30,132
composite factor = 2 × 32 × 5 × 11 × 31 = 30,690
composite factor = 22 × 36 × 11 = 32,076
composite factor = 23 × 33 × 5 × 31 = 33,480
composite factor = 23 × 34 × 5 × 11 = 35,640
composite factor = 22 × 33 × 11 × 31 = 36,828
composite factor = 35 × 5 × 31 = 37,665
composite factor = 36 × 5 × 11 = 40,095
composite factor = 23 × 3 × 5 × 11 × 31 = 40,920
composite factor = 22 × 37 × 5 = 43,740
composite factor = 2 × 36 × 31 = 45,198
composite factor = 33 × 5 × 11 × 31 = 46,035
composite factor = 2 × 37 × 11 = 48,114
composite factor = 22 × 34 × 5 × 31 = 50,220
composite factor = 22 × 35 × 5 × 11 = 53,460
composite factor = 2 × 34 × 11 × 31 = 55,242
composite factor = 23 × 35 × 31 = 60,264
composite factor = 22 × 32 × 5 × 11 × 31 = 61,380
composite factor = 23 × 36 × 11 = 64,152
composite factor = 37 × 31 = 67,797
composite factor = 23 × 33 × 11 × 31 = 73,656
composite factor = 2 × 35 × 5 × 31 = 75,330
composite factor = 2 × 36 × 5 × 11 = 80,190
composite factor = 35 × 11 × 31 = 82,863
composite factor = 23 × 37 × 5 = 87,480
composite factor = 22 × 36 × 31 = 90,396
composite factor = 2 × 33 × 5 × 11 × 31 = 92,070
composite factor = 22 × 37 × 11 = 96,228
composite factor = 23 × 34 × 5 × 31 = 100,440
composite factor = 23 × 35 × 5 × 11 = 106,920
composite factor = 22 × 34 × 11 × 31 = 110,484
composite factor = 36 × 5 × 31 = 112,995
composite factor = 37 × 5 × 11 = 120,285
composite factor = 23 × 32 × 5 × 11 × 31 = 122,760
composite factor = 2 × 37 × 31 = 135,594
composite factor = 34 × 5 × 11 × 31 = 138,105
composite factor = 22 × 35 × 5 × 31 = 150,660
composite factor = 22 × 36 × 5 × 11 = 160,380
composite factor = 2 × 35 × 11 × 31 = 165,726
composite factor = 23 × 36 × 31 = 180,792
composite factor = 22 × 33 × 5 × 11 × 31 = 184,140
composite factor = 23 × 37 × 11 = 192,456
composite factor = 23 × 34 × 11 × 31 = 220,968
composite factor = 2 × 36 × 5 × 31 = 225,990
composite factor = 2 × 37 × 5 × 11 = 240,570
composite factor = 36 × 11 × 31 = 248,589
composite factor = 22 × 37 × 31 = 271,188
composite factor = 2 × 34 × 5 × 11 × 31 = 276,210
composite factor = 23 × 35 × 5 × 31 = 301,320
composite factor = 23 × 36 × 5 × 11 = 320,760
composite factor = 22 × 35 × 11 × 31 = 331,452
composite factor = 37 × 5 × 31 = 338,985
composite factor = 23 × 33 × 5 × 11 × 31 = 368,280
composite factor = 35 × 5 × 11 × 31 = 414,315
composite factor = 22 × 36 × 5 × 31 = 451,980
composite factor = 22 × 37 × 5 × 11 = 481,140
composite factor = 2 × 36 × 11 × 31 = 497,178
composite factor = 23 × 37 × 31 = 542,376
composite factor = 22 × 34 × 5 × 11 × 31 = 552,420
composite factor = 23 × 35 × 11 × 31 = 662,904
composite factor = 2 × 37 × 5 × 31 = 677,970
composite factor = 37 × 11 × 31 = 745,767
composite factor = 2 × 35 × 5 × 11 × 31 = 828,630
composite factor = 23 × 36 × 5 × 31 = 903,960
composite factor = 23 × 37 × 5 × 11 = 962,280
composite factor = 22 × 36 × 11 × 31 = 994,356
composite factor = 23 × 34 × 5 × 11 × 31 = 1,104,840
composite factor = 36 × 5 × 11 × 31 = 1,242,945
composite factor = 22 × 37 × 5 × 31 = 1,355,940
composite factor = 2 × 37 × 11 × 31 = 1,491,534
composite factor = 22 × 35 × 5 × 11 × 31 = 1,657,260
composite factor = 23 × 36 × 11 × 31 = 1,988,712
composite factor = 2 × 36 × 5 × 11 × 31 = 2,485,890
composite factor = 23 × 37 × 5 × 31 = 2,711,880
composite factor = 22 × 37 × 11 × 31 = 2,983,068
composite factor = 23 × 35 × 5 × 11 × 31 = 3,314,520
composite factor = 37 × 5 × 11 × 31 = 3,728,835
composite factor = 22 × 36 × 5 × 11 × 31 = 4,971,780
composite factor = 23 × 37 × 11 × 31 = 5,966,136
composite factor = 2 × 37 × 5 × 11 × 31 = 7,457,670
composite factor = 23 × 36 × 5 × 11 × 31 = 9,943,560
composite factor = 22 × 37 × 5 × 11 × 31 = 14,915,340
composite factor = 23 × 37 × 5 × 11 × 31 = 29,830,680
256 factors (divisors)

What times what is 29,830,680?
What number multiplied by what number equals 29,830,680?

All the combinations of any two natural numbers whose product equals 29,830,680.

1 × 29,830,680 = 29,830,680
2 × 14,915,340 = 29,830,680
3 × 9,943,560 = 29,830,680
4 × 7,457,670 = 29,830,680
5 × 5,966,136 = 29,830,680
6 × 4,971,780 = 29,830,680
8 × 3,728,835 = 29,830,680
9 × 3,314,520 = 29,830,680
10 × 2,983,068 = 29,830,680
11 × 2,711,880 = 29,830,680
12 × 2,485,890 = 29,830,680
15 × 1,988,712 = 29,830,680
18 × 1,657,260 = 29,830,680
20 × 1,491,534 = 29,830,680
22 × 1,355,940 = 29,830,680
24 × 1,242,945 = 29,830,680
27 × 1,104,840 = 29,830,680
30 × 994,356 = 29,830,680
31 × 962,280 = 29,830,680
33 × 903,960 = 29,830,680
36 × 828,630 = 29,830,680
40 × 745,767 = 29,830,680
44 × 677,970 = 29,830,680
45 × 662,904 = 29,830,680
54 × 552,420 = 29,830,680
55 × 542,376 = 29,830,680
60 × 497,178 = 29,830,680
62 × 481,140 = 29,830,680
66 × 451,980 = 29,830,680
72 × 414,315 = 29,830,680
81 × 368,280 = 29,830,680
88 × 338,985 = 29,830,680
90 × 331,452 = 29,830,680
93 × 320,760 = 29,830,680
99 × 301,320 = 29,830,680
108 × 276,210 = 29,830,680
110 × 271,188 = 29,830,680
120 × 248,589 = 29,830,680
124 × 240,570 = 29,830,680
132 × 225,990 = 29,830,680
135 × 220,968 = 29,830,680
155 × 192,456 = 29,830,680
162 × 184,140 = 29,830,680
165 × 180,792 = 29,830,680
180 × 165,726 = 29,830,680
186 × 160,380 = 29,830,680
198 × 150,660 = 29,830,680
216 × 138,105 = 29,830,680
220 × 135,594 = 29,830,680
243 × 122,760 = 29,830,680
248 × 120,285 = 29,830,680
264 × 112,995 = 29,830,680
270 × 110,484 = 29,830,680
279 × 106,920 = 29,830,680
297 × 100,440 = 29,830,680
310 × 96,228 = 29,830,680
324 × 92,070 = 29,830,680
330 × 90,396 = 29,830,680
341 × 87,480 = 29,830,680
360 × 82,863 = 29,830,680
372 × 80,190 = 29,830,680
396 × 75,330 = 29,830,680
405 × 73,656 = 29,830,680
440 × 67,797 = 29,830,680
465 × 64,152 = 29,830,680
486 × 61,380 = 29,830,680
495 × 60,264 = 29,830,680
540 × 55,242 = 29,830,680
558 × 53,460 = 29,830,680
594 × 50,220 = 29,830,680
620 × 48,114 = 29,830,680
648 × 46,035 = 29,830,680
660 × 45,198 = 29,830,680
682 × 43,740 = 29,830,680
729 × 40,920 = 29,830,680
744 × 40,095 = 29,830,680
792 × 37,665 = 29,830,680
810 × 36,828 = 29,830,680
837 × 35,640 = 29,830,680
891 × 33,480 = 29,830,680
930 × 32,076 = 29,830,680
972 × 30,690 = 29,830,680
990 × 30,132 = 29,830,680
1,023 × 29,160 = 29,830,680
1,080 × 27,621 = 29,830,680
1,116 × 26,730 = 29,830,680
1,188 × 25,110 = 29,830,680
1,215 × 24,552 = 29,830,680
1,240 × 24,057 = 29,830,680
1,320 × 22,599 = 29,830,680
1,364 × 21,870 = 29,830,680
1,395 × 21,384 = 29,830,680
1,458 × 20,460 = 29,830,680
1,485 × 20,088 = 29,830,680
1,620 × 18,414 = 29,830,680
1,674 × 17,820 = 29,830,680
1,705 × 17,496 = 29,830,680
1,782 × 16,740 = 29,830,680
1,860 × 16,038 = 29,830,680
1,944 × 15,345 = 29,830,680
1,980 × 15,066 = 29,830,680
2,046 × 14,580 = 29,830,680
2,187 × 13,640 = 29,830,680
2,232 × 13,365 = 29,830,680
2,376 × 12,555 = 29,830,680
2,430 × 12,276 = 29,830,680
2,511 × 11,880 = 29,830,680
2,673 × 11,160 = 29,830,680
2,728 × 10,935 = 29,830,680
2,790 × 10,692 = 29,830,680
2,916 × 10,230 = 29,830,680
2,970 × 10,044 = 29,830,680
3,069 × 9,720 = 29,830,680
3,240 × 9,207 = 29,830,680
3,348 × 8,910 = 29,830,680
3,410 × 8,748 = 29,830,680
3,564 × 8,370 = 29,830,680
3,645 × 8,184 = 29,830,680
3,720 × 8,019 = 29,830,680
3,960 × 7,533 = 29,830,680
4,092 × 7,290 = 29,830,680
4,185 × 7,128 = 29,830,680
4,374 × 6,820 = 29,830,680
4,455 × 6,696 = 29,830,680
4,860 × 6,138 = 29,830,680
5,022 × 5,940 = 29,830,680
5,115 × 5,832 = 29,830,680
5,346 × 5,580 = 29,830,680
128 unique multiplications

The final answer:
(scroll down)


29,830,680 has 256 factors (divisors):
1; 2; 3; 4; 5; 6; 8; 9; 10; 11; 12; 15; 18; 20; 22; 24; 27; 30; 31; 33; 36; 40; 44; 45; 54; 55; 60; 62; 66; 72; 81; 88; 90; 93; 99; 108; 110; 120; 124; 132; 135; 155; 162; 165; 180; 186; 198; 216; 220; 243; 248; 264; 270; 279; 297; 310; 324; 330; 341; 360; 372; 396; 405; 440; 465; 486; 495; 540; 558; 594; 620; 648; 660; 682; 729; 744; 792; 810; 837; 891; 930; 972; 990; 1,023; 1,080; 1,116; 1,188; 1,215; 1,240; 1,320; 1,364; 1,395; 1,458; 1,485; 1,620; 1,674; 1,705; 1,782; 1,860; 1,944; 1,980; 2,046; 2,187; 2,232; 2,376; 2,430; 2,511; 2,673; 2,728; 2,790; 2,916; 2,970; 3,069; 3,240; 3,348; 3,410; 3,564; 3,645; 3,720; 3,960; 4,092; 4,185; 4,374; 4,455; 4,860; 5,022; 5,115; 5,346; 5,580; 5,832; 5,940; 6,138; 6,696; 6,820; 7,128; 7,290; 7,533; 8,019; 8,184; 8,370; 8,748; 8,910; 9,207; 9,720; 10,044; 10,230; 10,692; 10,935; 11,160; 11,880; 12,276; 12,555; 13,365; 13,640; 14,580; 15,066; 15,345; 16,038; 16,740; 17,496; 17,820; 18,414; 20,088; 20,460; 21,384; 21,870; 22,599; 24,057; 24,552; 25,110; 26,730; 27,621; 29,160; 30,132; 30,690; 32,076; 33,480; 35,640; 36,828; 37,665; 40,095; 40,920; 43,740; 45,198; 46,035; 48,114; 50,220; 53,460; 55,242; 60,264; 61,380; 64,152; 67,797; 73,656; 75,330; 80,190; 82,863; 87,480; 90,396; 92,070; 96,228; 100,440; 106,920; 110,484; 112,995; 120,285; 122,760; 135,594; 138,105; 150,660; 160,380; 165,726; 180,792; 184,140; 192,456; 220,968; 225,990; 240,570; 248,589; 271,188; 276,210; 301,320; 320,760; 331,452; 338,985; 368,280; 414,315; 451,980; 481,140; 497,178; 542,376; 552,420; 662,904; 677,970; 745,767; 828,630; 903,960; 962,280; 994,356; 1,104,840; 1,242,945; 1,355,940; 1,491,534; 1,657,260; 1,988,712; 2,485,890; 2,711,880; 2,983,068; 3,314,520; 3,728,835; 4,971,780; 5,966,136; 7,457,670; 9,943,560; 14,915,340 and 29,830,680
out of which 5 prime factors: 2; 3; 5; 11 and 31.
Numbers other than 1 that are not prime factors are composite factors (divisors).
29,830,680 and 1 are sometimes called improper factors, the others are called proper factors (proper divisors).

  • A quick way to find the factors (the divisors) of a number is to break it down into prime factors.
  • Then multiply the prime factors and their exponents, if any, in all their different combinations.

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Factors (divisors), common factors (common divisors), the greatest common factor, GCF (also called the greatest common divisor, GCD, or the highest common factor, HCF)

  • If the number "t" is a factor (divisor) of the number "a" then in the prime factorization of "t" we will only encounter prime factors that also occur in the prime factorization of "a".
  • If there are exponents involved, the maximum value of an exponent for any base of a power that is found in the prime factorization of "t" (powers, or multiplicities) is at most equal to the exponent of the same base that is involved in the prime factorization of "a".
  • Hint: 23 = 2 × 2 × 2 = 8. 2 is called the base and 3 is the exponent. 23 is the power and 8 is the value of the power. We sometimes say that the number 2 is raised to the power of 3.
  • For example, 12 is a factor (divisor) of 120 - the remainder is zero when dividing 120 by 12.
  • Let's look at the prime factorization of both numbers and notice the bases and the exponents that occur in the prime factorization of both numbers:
  • 12 = 2 × 2 × 3 = 22 × 3
  • 120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5
  • 120 contains all the prime factors of 12, and all its bases' exponents are higher than those of 12.
  • If "t" is a common factor (divisor) of "a" and "b", then the prime factorization of "t" contains only the common prime factors involved in the prime factorizations of both "a" and "b".
  • If there are exponents involved, the maximum value of an exponent for any base of a power that is found in the prime factorization of "t" is at most equal to the minimum of the exponents of the same base that is involved in the prime factorization of both "a" and "b".
  • For example, 12 is the common factor of 48 and 360.
  • The remainder is zero when dividing either 48 or 360 by 12.
  • Here there are the prime factorizations of the three numbers, 12, 48 and 360:
  • 12 = 22 × 3
  • 48 = 24 × 3
  • 360 = 23 × 32 × 5
  • Please note that 48 and 360 have more factors (divisors): 2, 3, 4, 6, 8, 12, 24. Among them, 24 is the greatest common factor, GCF (or the greatest common divisor, GCD, or the highest common factor, HCF) of 48 and 360.
  • The greatest common factor, GCF, of two numbers, "a" and "b", is the product of all the common prime factors involved in the prime factorizations of both "a" and "b", taken by the lowest exponents.
  • Based on this rule it is calculated the greatest common factor, GCF, (or the greatest common divisor GCD, HCF) of several numbers, as shown in the example below...
  • GCF, GCD (1,260; 3,024; 5,544) = ?
  • 1,260 = 22 × 32
  • 3,024 = 24 × 32 × 7
  • 5,544 = 23 × 32 × 7 × 11
  • The common prime factors are:
  • 2 - its lowest exponent (multiplicity) is: min.(2; 3; 4) = 2
  • 3 - its lowest exponent (multiplicity) is: min.(2; 2; 2) = 2
  • GCF, GCD (1,260; 3,024; 5,544) = 22 × 32 = 252
  • Coprime numbers:
  • If two numbers "a" and "b" have no other common factors (divisors) than 1, gfc, gcd, hcf (a; b) = 1, then the numbers "a" and "b" are called coprime (or relatively prime).
  • Factors of the GCF
  • If "a" and "b" are not coprime, then every common factor (divisor) of "a" and "b" is a also a factor (divisor) of the greatest common factor, GCF (greatest common divisor, GCD, highest common factor, HCF) of "a" and "b".