Factors of 338,170,320. Calculator of Proper, Improper, Prime and Compound Divisors, if Any

All the factors (divisors) of the number 338,170,320. Connection with the prime factorization of the number

To find all the divisors of the number 338,170,320:

  • 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 338,170,320:

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


338,170,320 = 24 × 32 × 5 × 31 × 109 × 139
338,170,320 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 = (4 + 1) × (2 + 1) × (1 + 1) × (1 + 1) × (1 + 1) × (1 + 1) = 5 × 3 × 2 × 2 × 2 × 2 = 240

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

2. Multiply the prime factors of the number 338,170,320

  • 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
composite factor = 22 × 3 = 12
composite factor = 3 × 5 = 15
composite factor = 24 = 16
composite factor = 2 × 32 = 18
composite factor = 22 × 5 = 20
composite factor = 23 × 3 = 24
composite factor = 2 × 3 × 5 = 30
prime factor = 31
composite factor = 22 × 32 = 36
composite factor = 23 × 5 = 40
composite factor = 32 × 5 = 45
composite factor = 24 × 3 = 48
composite factor = 22 × 3 × 5 = 60
composite factor = 2 × 31 = 62
composite factor = 23 × 32 = 72
composite factor = 24 × 5 = 80
composite factor = 2 × 32 × 5 = 90
composite factor = 3 × 31 = 93
prime factor = 109
composite factor = 23 × 3 × 5 = 120
composite factor = 22 × 31 = 124
prime factor = 139
composite factor = 24 × 32 = 144
composite factor = 5 × 31 = 155
composite factor = 22 × 32 × 5 = 180
composite factor = 2 × 3 × 31 = 186
composite factor = 2 × 109 = 218
composite factor = 24 × 3 × 5 = 240
composite factor = 23 × 31 = 248
composite factor = 2 × 139 = 278
composite factor = 32 × 31 = 279
composite factor = 2 × 5 × 31 = 310
composite factor = 3 × 109 = 327
composite factor = 23 × 32 × 5 = 360
composite factor = 22 × 3 × 31 = 372
composite factor = 3 × 139 = 417
composite factor = 22 × 109 = 436
composite factor = 3 × 5 × 31 = 465
composite factor = 24 × 31 = 496
composite factor = 5 × 109 = 545
composite factor = 22 × 139 = 556
composite factor = 2 × 32 × 31 = 558
composite factor = 22 × 5 × 31 = 620
composite factor = 2 × 3 × 109 = 654
composite factor = 5 × 139 = 695
composite factor = 24 × 32 × 5 = 720
composite factor = 23 × 3 × 31 = 744
composite factor = 2 × 3 × 139 = 834
composite factor = 23 × 109 = 872
composite factor = 2 × 3 × 5 × 31 = 930
composite factor = 32 × 109 = 981
composite factor = 2 × 5 × 109 = 1,090
composite factor = 23 × 139 = 1,112
composite factor = 22 × 32 × 31 = 1,116
composite factor = 23 × 5 × 31 = 1,240
composite factor = 32 × 139 = 1,251
composite factor = 22 × 3 × 109 = 1,308
composite factor = 2 × 5 × 139 = 1,390
composite factor = 32 × 5 × 31 = 1,395
composite factor = 24 × 3 × 31 = 1,488
composite factor = 3 × 5 × 109 = 1,635
composite factor = 22 × 3 × 139 = 1,668
composite factor = 24 × 109 = 1,744
composite factor = 22 × 3 × 5 × 31 = 1,860
composite factor = 2 × 32 × 109 = 1,962
composite factor = 3 × 5 × 139 = 2,085
composite factor = 22 × 5 × 109 = 2,180
composite factor = 24 × 139 = 2,224
composite factor = 23 × 32 × 31 = 2,232
composite factor = 24 × 5 × 31 = 2,480
composite factor = 2 × 32 × 139 = 2,502
composite factor = 23 × 3 × 109 = 2,616
composite factor = 22 × 5 × 139 = 2,780
composite factor = 2 × 32 × 5 × 31 = 2,790
composite factor = 2 × 3 × 5 × 109 = 3,270
composite factor = 23 × 3 × 139 = 3,336
composite factor = 31 × 109 = 3,379
composite factor = 23 × 3 × 5 × 31 = 3,720
composite factor = 22 × 32 × 109 = 3,924
composite factor = 2 × 3 × 5 × 139 = 4,170
composite factor = 31 × 139 = 4,309
composite factor = 23 × 5 × 109 = 4,360
composite factor = 24 × 32 × 31 = 4,464
composite factor = 32 × 5 × 109 = 4,905
composite factor = 22 × 32 × 139 = 5,004
composite factor = 24 × 3 × 109 = 5,232
composite factor = 23 × 5 × 139 = 5,560
composite factor = 22 × 32 × 5 × 31 = 5,580
composite factor = 32 × 5 × 139 = 6,255
composite factor = 22 × 3 × 5 × 109 = 6,540
composite factor = 24 × 3 × 139 = 6,672
composite factor = 2 × 31 × 109 = 6,758
composite factor = 24 × 3 × 5 × 31 = 7,440
composite factor = 23 × 32 × 109 = 7,848
composite factor = 22 × 3 × 5 × 139 = 8,340
composite factor = 2 × 31 × 139 = 8,618
composite factor = 24 × 5 × 109 = 8,720
composite factor = 2 × 32 × 5 × 109 = 9,810
composite factor = 23 × 32 × 139 = 10,008
composite factor = 3 × 31 × 109 = 10,137
composite factor = 24 × 5 × 139 = 11,120
composite factor = 23 × 32 × 5 × 31 = 11,160
composite factor = 2 × 32 × 5 × 139 = 12,510
composite factor = 3 × 31 × 139 = 12,927
composite factor = 23 × 3 × 5 × 109 = 13,080
composite factor = 22 × 31 × 109 = 13,516
composite factor = 109 × 139 = 15,151
composite factor = 24 × 32 × 109 = 15,696
composite factor = 23 × 3 × 5 × 139 = 16,680
composite factor = 5 × 31 × 109 = 16,895
composite factor = 22 × 31 × 139 = 17,236
This list continues below...

... This list continues from above
composite factor = 22 × 32 × 5 × 109 = 19,620
composite factor = 24 × 32 × 139 = 20,016
composite factor = 2 × 3 × 31 × 109 = 20,274
composite factor = 5 × 31 × 139 = 21,545
composite factor = 24 × 32 × 5 × 31 = 22,320
composite factor = 22 × 32 × 5 × 139 = 25,020
composite factor = 2 × 3 × 31 × 139 = 25,854
composite factor = 24 × 3 × 5 × 109 = 26,160
composite factor = 23 × 31 × 109 = 27,032
composite factor = 2 × 109 × 139 = 30,302
composite factor = 32 × 31 × 109 = 30,411
composite factor = 24 × 3 × 5 × 139 = 33,360
composite factor = 2 × 5 × 31 × 109 = 33,790
composite factor = 23 × 31 × 139 = 34,472
composite factor = 32 × 31 × 139 = 38,781
composite factor = 23 × 32 × 5 × 109 = 39,240
composite factor = 22 × 3 × 31 × 109 = 40,548
composite factor = 2 × 5 × 31 × 139 = 43,090
composite factor = 3 × 109 × 139 = 45,453
composite factor = 23 × 32 × 5 × 139 = 50,040
composite factor = 3 × 5 × 31 × 109 = 50,685
composite factor = 22 × 3 × 31 × 139 = 51,708
composite factor = 24 × 31 × 109 = 54,064
composite factor = 22 × 109 × 139 = 60,604
composite factor = 2 × 32 × 31 × 109 = 60,822
composite factor = 3 × 5 × 31 × 139 = 64,635
composite factor = 22 × 5 × 31 × 109 = 67,580
composite factor = 24 × 31 × 139 = 68,944
composite factor = 5 × 109 × 139 = 75,755
composite factor = 2 × 32 × 31 × 139 = 77,562
composite factor = 24 × 32 × 5 × 109 = 78,480
composite factor = 23 × 3 × 31 × 109 = 81,096
composite factor = 22 × 5 × 31 × 139 = 86,180
composite factor = 2 × 3 × 109 × 139 = 90,906
composite factor = 24 × 32 × 5 × 139 = 100,080
composite factor = 2 × 3 × 5 × 31 × 109 = 101,370
composite factor = 23 × 3 × 31 × 139 = 103,416
composite factor = 23 × 109 × 139 = 121,208
composite factor = 22 × 32 × 31 × 109 = 121,644
composite factor = 2 × 3 × 5 × 31 × 139 = 129,270
composite factor = 23 × 5 × 31 × 109 = 135,160
composite factor = 32 × 109 × 139 = 136,359
composite factor = 2 × 5 × 109 × 139 = 151,510
composite factor = 32 × 5 × 31 × 109 = 152,055
composite factor = 22 × 32 × 31 × 139 = 155,124
composite factor = 24 × 3 × 31 × 109 = 162,192
composite factor = 23 × 5 × 31 × 139 = 172,360
composite factor = 22 × 3 × 109 × 139 = 181,812
composite factor = 32 × 5 × 31 × 139 = 193,905
composite factor = 22 × 3 × 5 × 31 × 109 = 202,740
composite factor = 24 × 3 × 31 × 139 = 206,832
composite factor = 3 × 5 × 109 × 139 = 227,265
composite factor = 24 × 109 × 139 = 242,416
composite factor = 23 × 32 × 31 × 109 = 243,288
composite factor = 22 × 3 × 5 × 31 × 139 = 258,540
composite factor = 24 × 5 × 31 × 109 = 270,320
composite factor = 2 × 32 × 109 × 139 = 272,718
composite factor = 22 × 5 × 109 × 139 = 303,020
composite factor = 2 × 32 × 5 × 31 × 109 = 304,110
composite factor = 23 × 32 × 31 × 139 = 310,248
composite factor = 24 × 5 × 31 × 139 = 344,720
composite factor = 23 × 3 × 109 × 139 = 363,624
composite factor = 2 × 32 × 5 × 31 × 139 = 387,810
composite factor = 23 × 3 × 5 × 31 × 109 = 405,480
composite factor = 2 × 3 × 5 × 109 × 139 = 454,530
composite factor = 31 × 109 × 139 = 469,681
composite factor = 24 × 32 × 31 × 109 = 486,576
composite factor = 23 × 3 × 5 × 31 × 139 = 517,080
composite factor = 22 × 32 × 109 × 139 = 545,436
composite factor = 23 × 5 × 109 × 139 = 606,040
composite factor = 22 × 32 × 5 × 31 × 109 = 608,220
composite factor = 24 × 32 × 31 × 139 = 620,496
composite factor = 32 × 5 × 109 × 139 = 681,795
composite factor = 24 × 3 × 109 × 139 = 727,248
composite factor = 22 × 32 × 5 × 31 × 139 = 775,620
composite factor = 24 × 3 × 5 × 31 × 109 = 810,960
composite factor = 22 × 3 × 5 × 109 × 139 = 909,060
composite factor = 2 × 31 × 109 × 139 = 939,362
composite factor = 24 × 3 × 5 × 31 × 139 = 1,034,160
composite factor = 23 × 32 × 109 × 139 = 1,090,872
composite factor = 24 × 5 × 109 × 139 = 1,212,080
composite factor = 23 × 32 × 5 × 31 × 109 = 1,216,440
composite factor = 2 × 32 × 5 × 109 × 139 = 1,363,590
composite factor = 3 × 31 × 109 × 139 = 1,409,043
composite factor = 23 × 32 × 5 × 31 × 139 = 1,551,240
composite factor = 23 × 3 × 5 × 109 × 139 = 1,818,120
composite factor = 22 × 31 × 109 × 139 = 1,878,724
composite factor = 24 × 32 × 109 × 139 = 2,181,744
composite factor = 5 × 31 × 109 × 139 = 2,348,405
composite factor = 24 × 32 × 5 × 31 × 109 = 2,432,880
composite factor = 22 × 32 × 5 × 109 × 139 = 2,727,180
composite factor = 2 × 3 × 31 × 109 × 139 = 2,818,086
composite factor = 24 × 32 × 5 × 31 × 139 = 3,102,480
composite factor = 24 × 3 × 5 × 109 × 139 = 3,636,240
composite factor = 23 × 31 × 109 × 139 = 3,757,448
composite factor = 32 × 31 × 109 × 139 = 4,227,129
composite factor = 2 × 5 × 31 × 109 × 139 = 4,696,810
composite factor = 23 × 32 × 5 × 109 × 139 = 5,454,360
composite factor = 22 × 3 × 31 × 109 × 139 = 5,636,172
composite factor = 3 × 5 × 31 × 109 × 139 = 7,045,215
composite factor = 24 × 31 × 109 × 139 = 7,514,896
composite factor = 2 × 32 × 31 × 109 × 139 = 8,454,258
composite factor = 22 × 5 × 31 × 109 × 139 = 9,393,620
composite factor = 24 × 32 × 5 × 109 × 139 = 10,908,720
composite factor = 23 × 3 × 31 × 109 × 139 = 11,272,344
composite factor = 2 × 3 × 5 × 31 × 109 × 139 = 14,090,430
composite factor = 22 × 32 × 31 × 109 × 139 = 16,908,516
composite factor = 23 × 5 × 31 × 109 × 139 = 18,787,240
composite factor = 32 × 5 × 31 × 109 × 139 = 21,135,645
composite factor = 24 × 3 × 31 × 109 × 139 = 22,544,688
composite factor = 22 × 3 × 5 × 31 × 109 × 139 = 28,180,860
composite factor = 23 × 32 × 31 × 109 × 139 = 33,817,032
composite factor = 24 × 5 × 31 × 109 × 139 = 37,574,480
composite factor = 2 × 32 × 5 × 31 × 109 × 139 = 42,271,290
composite factor = 23 × 3 × 5 × 31 × 109 × 139 = 56,361,720
composite factor = 24 × 32 × 31 × 109 × 139 = 67,634,064
composite factor = 22 × 32 × 5 × 31 × 109 × 139 = 84,542,580
composite factor = 24 × 3 × 5 × 31 × 109 × 139 = 112,723,440
composite factor = 23 × 32 × 5 × 31 × 109 × 139 = 169,085,160
composite factor = 24 × 32 × 5 × 31 × 109 × 139 = 338,170,320
240 factors (divisors)

What times what is 338,170,320?
What number multiplied by what number equals 338,170,320?

All the combinations of any two natural numbers whose product equals 338,170,320.

1 × 338,170,320 = 338,170,320
2 × 169,085,160 = 338,170,320
3 × 112,723,440 = 338,170,320
4 × 84,542,580 = 338,170,320
5 × 67,634,064 = 338,170,320
6 × 56,361,720 = 338,170,320
8 × 42,271,290 = 338,170,320
9 × 37,574,480 = 338,170,320
10 × 33,817,032 = 338,170,320
12 × 28,180,860 = 338,170,320
15 × 22,544,688 = 338,170,320
16 × 21,135,645 = 338,170,320
18 × 18,787,240 = 338,170,320
20 × 16,908,516 = 338,170,320
24 × 14,090,430 = 338,170,320
30 × 11,272,344 = 338,170,320
31 × 10,908,720 = 338,170,320
36 × 9,393,620 = 338,170,320
40 × 8,454,258 = 338,170,320
45 × 7,514,896 = 338,170,320
48 × 7,045,215 = 338,170,320
60 × 5,636,172 = 338,170,320
62 × 5,454,360 = 338,170,320
72 × 4,696,810 = 338,170,320
80 × 4,227,129 = 338,170,320
90 × 3,757,448 = 338,170,320
93 × 3,636,240 = 338,170,320
109 × 3,102,480 = 338,170,320
120 × 2,818,086 = 338,170,320
124 × 2,727,180 = 338,170,320
139 × 2,432,880 = 338,170,320
144 × 2,348,405 = 338,170,320
155 × 2,181,744 = 338,170,320
180 × 1,878,724 = 338,170,320
186 × 1,818,120 = 338,170,320
218 × 1,551,240 = 338,170,320
240 × 1,409,043 = 338,170,320
248 × 1,363,590 = 338,170,320
278 × 1,216,440 = 338,170,320
279 × 1,212,080 = 338,170,320
310 × 1,090,872 = 338,170,320
327 × 1,034,160 = 338,170,320
360 × 939,362 = 338,170,320
372 × 909,060 = 338,170,320
417 × 810,960 = 338,170,320
436 × 775,620 = 338,170,320
465 × 727,248 = 338,170,320
496 × 681,795 = 338,170,320
545 × 620,496 = 338,170,320
556 × 608,220 = 338,170,320
558 × 606,040 = 338,170,320
620 × 545,436 = 338,170,320
654 × 517,080 = 338,170,320
695 × 486,576 = 338,170,320
720 × 469,681 = 338,170,320
744 × 454,530 = 338,170,320
834 × 405,480 = 338,170,320
872 × 387,810 = 338,170,320
930 × 363,624 = 338,170,320
981 × 344,720 = 338,170,320
1,090 × 310,248 = 338,170,320
1,112 × 304,110 = 338,170,320
1,116 × 303,020 = 338,170,320
1,240 × 272,718 = 338,170,320
1,251 × 270,320 = 338,170,320
1,308 × 258,540 = 338,170,320
1,390 × 243,288 = 338,170,320
1,395 × 242,416 = 338,170,320
1,488 × 227,265 = 338,170,320
1,635 × 206,832 = 338,170,320
1,668 × 202,740 = 338,170,320
1,744 × 193,905 = 338,170,320
1,860 × 181,812 = 338,170,320
1,962 × 172,360 = 338,170,320
2,085 × 162,192 = 338,170,320
2,180 × 155,124 = 338,170,320
2,224 × 152,055 = 338,170,320
2,232 × 151,510 = 338,170,320
2,480 × 136,359 = 338,170,320
2,502 × 135,160 = 338,170,320
2,616 × 129,270 = 338,170,320
2,780 × 121,644 = 338,170,320
2,790 × 121,208 = 338,170,320
3,270 × 103,416 = 338,170,320
3,336 × 101,370 = 338,170,320
3,379 × 100,080 = 338,170,320
3,720 × 90,906 = 338,170,320
3,924 × 86,180 = 338,170,320
4,170 × 81,096 = 338,170,320
4,309 × 78,480 = 338,170,320
4,360 × 77,562 = 338,170,320
4,464 × 75,755 = 338,170,320
4,905 × 68,944 = 338,170,320
5,004 × 67,580 = 338,170,320
5,232 × 64,635 = 338,170,320
5,560 × 60,822 = 338,170,320
5,580 × 60,604 = 338,170,320
6,255 × 54,064 = 338,170,320
6,540 × 51,708 = 338,170,320
6,672 × 50,685 = 338,170,320
6,758 × 50,040 = 338,170,320
7,440 × 45,453 = 338,170,320
7,848 × 43,090 = 338,170,320
8,340 × 40,548 = 338,170,320
8,618 × 39,240 = 338,170,320
8,720 × 38,781 = 338,170,320
9,810 × 34,472 = 338,170,320
10,008 × 33,790 = 338,170,320
10,137 × 33,360 = 338,170,320
11,120 × 30,411 = 338,170,320
11,160 × 30,302 = 338,170,320
12,510 × 27,032 = 338,170,320
12,927 × 26,160 = 338,170,320
13,080 × 25,854 = 338,170,320
13,516 × 25,020 = 338,170,320
15,151 × 22,320 = 338,170,320
15,696 × 21,545 = 338,170,320
16,680 × 20,274 = 338,170,320
16,895 × 20,016 = 338,170,320
17,236 × 19,620 = 338,170,320
120 unique multiplications

The final answer:
(scroll down)


338,170,320 has 240 factors (divisors):
1; 2; 3; 4; 5; 6; 8; 9; 10; 12; 15; 16; 18; 20; 24; 30; 31; 36; 40; 45; 48; 60; 62; 72; 80; 90; 93; 109; 120; 124; 139; 144; 155; 180; 186; 218; 240; 248; 278; 279; 310; 327; 360; 372; 417; 436; 465; 496; 545; 556; 558; 620; 654; 695; 720; 744; 834; 872; 930; 981; 1,090; 1,112; 1,116; 1,240; 1,251; 1,308; 1,390; 1,395; 1,488; 1,635; 1,668; 1,744; 1,860; 1,962; 2,085; 2,180; 2,224; 2,232; 2,480; 2,502; 2,616; 2,780; 2,790; 3,270; 3,336; 3,379; 3,720; 3,924; 4,170; 4,309; 4,360; 4,464; 4,905; 5,004; 5,232; 5,560; 5,580; 6,255; 6,540; 6,672; 6,758; 7,440; 7,848; 8,340; 8,618; 8,720; 9,810; 10,008; 10,137; 11,120; 11,160; 12,510; 12,927; 13,080; 13,516; 15,151; 15,696; 16,680; 16,895; 17,236; 19,620; 20,016; 20,274; 21,545; 22,320; 25,020; 25,854; 26,160; 27,032; 30,302; 30,411; 33,360; 33,790; 34,472; 38,781; 39,240; 40,548; 43,090; 45,453; 50,040; 50,685; 51,708; 54,064; 60,604; 60,822; 64,635; 67,580; 68,944; 75,755; 77,562; 78,480; 81,096; 86,180; 90,906; 100,080; 101,370; 103,416; 121,208; 121,644; 129,270; 135,160; 136,359; 151,510; 152,055; 155,124; 162,192; 172,360; 181,812; 193,905; 202,740; 206,832; 227,265; 242,416; 243,288; 258,540; 270,320; 272,718; 303,020; 304,110; 310,248; 344,720; 363,624; 387,810; 405,480; 454,530; 469,681; 486,576; 517,080; 545,436; 606,040; 608,220; 620,496; 681,795; 727,248; 775,620; 810,960; 909,060; 939,362; 1,034,160; 1,090,872; 1,212,080; 1,216,440; 1,363,590; 1,409,043; 1,551,240; 1,818,120; 1,878,724; 2,181,744; 2,348,405; 2,432,880; 2,727,180; 2,818,086; 3,102,480; 3,636,240; 3,757,448; 4,227,129; 4,696,810; 5,454,360; 5,636,172; 7,045,215; 7,514,896; 8,454,258; 9,393,620; 10,908,720; 11,272,344; 14,090,430; 16,908,516; 18,787,240; 21,135,645; 22,544,688; 28,180,860; 33,817,032; 37,574,480; 42,271,290; 56,361,720; 67,634,064; 84,542,580; 112,723,440; 169,085,160 and 338,170,320
out of which 6 prime factors: 2; 3; 5; 31; 109 and 139.
Numbers other than 1 that are not prime factors are composite factors (divisors).
338,170,320 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".