Factors of 85,642,800. Calculator of Proper, Improper, Prime and Compound Divisors, if Any

All the factors (divisors) of the number 85,642,800. Connection with the prime factorization of the number

To find all the divisors of the number 85,642,800:

  • 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 85,642,800:

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


85,642,800 = 24 × 3 × 52 × 23 × 29 × 107
85,642,800 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) × (1 + 1) × (2 + 1) × (1 + 1) × (1 + 1) × (1 + 1) = 5 × 2 × 3 × 2 × 2 × 2 = 240

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

2. Multiply the prime factors of the number 85,642,800

  • 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 = 2 × 5 = 10
composite factor = 22 × 3 = 12
composite factor = 3 × 5 = 15
composite factor = 24 = 16
composite factor = 22 × 5 = 20
prime factor = 23
composite factor = 23 × 3 = 24
composite factor = 52 = 25
prime factor = 29
composite factor = 2 × 3 × 5 = 30
composite factor = 23 × 5 = 40
composite factor = 2 × 23 = 46
composite factor = 24 × 3 = 48
composite factor = 2 × 52 = 50
composite factor = 2 × 29 = 58
composite factor = 22 × 3 × 5 = 60
composite factor = 3 × 23 = 69
composite factor = 3 × 52 = 75
composite factor = 24 × 5 = 80
composite factor = 3 × 29 = 87
composite factor = 22 × 23 = 92
composite factor = 22 × 52 = 100
prime factor = 107
composite factor = 5 × 23 = 115
composite factor = 22 × 29 = 116
composite factor = 23 × 3 × 5 = 120
composite factor = 2 × 3 × 23 = 138
composite factor = 5 × 29 = 145
composite factor = 2 × 3 × 52 = 150
composite factor = 2 × 3 × 29 = 174
composite factor = 23 × 23 = 184
composite factor = 23 × 52 = 200
composite factor = 2 × 107 = 214
composite factor = 2 × 5 × 23 = 230
composite factor = 23 × 29 = 232
composite factor = 24 × 3 × 5 = 240
composite factor = 22 × 3 × 23 = 276
composite factor = 2 × 5 × 29 = 290
composite factor = 22 × 3 × 52 = 300
composite factor = 3 × 107 = 321
composite factor = 3 × 5 × 23 = 345
composite factor = 22 × 3 × 29 = 348
composite factor = 24 × 23 = 368
composite factor = 24 × 52 = 400
composite factor = 22 × 107 = 428
composite factor = 3 × 5 × 29 = 435
composite factor = 22 × 5 × 23 = 460
composite factor = 24 × 29 = 464
composite factor = 5 × 107 = 535
composite factor = 23 × 3 × 23 = 552
composite factor = 52 × 23 = 575
composite factor = 22 × 5 × 29 = 580
composite factor = 23 × 3 × 52 = 600
composite factor = 2 × 3 × 107 = 642
composite factor = 23 × 29 = 667
composite factor = 2 × 3 × 5 × 23 = 690
composite factor = 23 × 3 × 29 = 696
composite factor = 52 × 29 = 725
composite factor = 23 × 107 = 856
composite factor = 2 × 3 × 5 × 29 = 870
composite factor = 23 × 5 × 23 = 920
composite factor = 2 × 5 × 107 = 1,070
composite factor = 24 × 3 × 23 = 1,104
composite factor = 2 × 52 × 23 = 1,150
composite factor = 23 × 5 × 29 = 1,160
composite factor = 24 × 3 × 52 = 1,200
composite factor = 22 × 3 × 107 = 1,284
composite factor = 2 × 23 × 29 = 1,334
composite factor = 22 × 3 × 5 × 23 = 1,380
composite factor = 24 × 3 × 29 = 1,392
composite factor = 2 × 52 × 29 = 1,450
composite factor = 3 × 5 × 107 = 1,605
composite factor = 24 × 107 = 1,712
composite factor = 3 × 52 × 23 = 1,725
composite factor = 22 × 3 × 5 × 29 = 1,740
composite factor = 24 × 5 × 23 = 1,840
composite factor = 3 × 23 × 29 = 2,001
composite factor = 22 × 5 × 107 = 2,140
composite factor = 3 × 52 × 29 = 2,175
composite factor = 22 × 52 × 23 = 2,300
composite factor = 24 × 5 × 29 = 2,320
composite factor = 23 × 107 = 2,461
composite factor = 23 × 3 × 107 = 2,568
composite factor = 22 × 23 × 29 = 2,668
composite factor = 52 × 107 = 2,675
composite factor = 23 × 3 × 5 × 23 = 2,760
composite factor = 22 × 52 × 29 = 2,900
composite factor = 29 × 107 = 3,103
composite factor = 2 × 3 × 5 × 107 = 3,210
composite factor = 5 × 23 × 29 = 3,335
composite factor = 2 × 3 × 52 × 23 = 3,450
composite factor = 23 × 3 × 5 × 29 = 3,480
composite factor = 2 × 3 × 23 × 29 = 4,002
composite factor = 23 × 5 × 107 = 4,280
composite factor = 2 × 3 × 52 × 29 = 4,350
composite factor = 23 × 52 × 23 = 4,600
composite factor = 2 × 23 × 107 = 4,922
composite factor = 24 × 3 × 107 = 5,136
composite factor = 23 × 23 × 29 = 5,336
composite factor = 2 × 52 × 107 = 5,350
composite factor = 24 × 3 × 5 × 23 = 5,520
composite factor = 23 × 52 × 29 = 5,800
composite factor = 2 × 29 × 107 = 6,206
composite factor = 22 × 3 × 5 × 107 = 6,420
composite factor = 2 × 5 × 23 × 29 = 6,670
composite factor = 22 × 3 × 52 × 23 = 6,900
composite factor = 24 × 3 × 5 × 29 = 6,960
composite factor = 3 × 23 × 107 = 7,383
composite factor = 22 × 3 × 23 × 29 = 8,004
composite factor = 3 × 52 × 107 = 8,025
composite factor = 24 × 5 × 107 = 8,560
composite factor = 22 × 3 × 52 × 29 = 8,700
composite factor = 24 × 52 × 23 = 9,200
This list continues below...

... This list continues from above
composite factor = 3 × 29 × 107 = 9,309
composite factor = 22 × 23 × 107 = 9,844
composite factor = 3 × 5 × 23 × 29 = 10,005
composite factor = 24 × 23 × 29 = 10,672
composite factor = 22 × 52 × 107 = 10,700
composite factor = 24 × 52 × 29 = 11,600
composite factor = 5 × 23 × 107 = 12,305
composite factor = 22 × 29 × 107 = 12,412
composite factor = 23 × 3 × 5 × 107 = 12,840
composite factor = 22 × 5 × 23 × 29 = 13,340
composite factor = 23 × 3 × 52 × 23 = 13,800
composite factor = 2 × 3 × 23 × 107 = 14,766
composite factor = 5 × 29 × 107 = 15,515
composite factor = 23 × 3 × 23 × 29 = 16,008
composite factor = 2 × 3 × 52 × 107 = 16,050
composite factor = 52 × 23 × 29 = 16,675
composite factor = 23 × 3 × 52 × 29 = 17,400
composite factor = 2 × 3 × 29 × 107 = 18,618
composite factor = 23 × 23 × 107 = 19,688
composite factor = 2 × 3 × 5 × 23 × 29 = 20,010
composite factor = 23 × 52 × 107 = 21,400
composite factor = 2 × 5 × 23 × 107 = 24,610
composite factor = 23 × 29 × 107 = 24,824
composite factor = 24 × 3 × 5 × 107 = 25,680
composite factor = 23 × 5 × 23 × 29 = 26,680
composite factor = 24 × 3 × 52 × 23 = 27,600
composite factor = 22 × 3 × 23 × 107 = 29,532
composite factor = 2 × 5 × 29 × 107 = 31,030
composite factor = 24 × 3 × 23 × 29 = 32,016
composite factor = 22 × 3 × 52 × 107 = 32,100
composite factor = 2 × 52 × 23 × 29 = 33,350
composite factor = 24 × 3 × 52 × 29 = 34,800
composite factor = 3 × 5 × 23 × 107 = 36,915
composite factor = 22 × 3 × 29 × 107 = 37,236
composite factor = 24 × 23 × 107 = 39,376
composite factor = 22 × 3 × 5 × 23 × 29 = 40,020
composite factor = 24 × 52 × 107 = 42,800
composite factor = 3 × 5 × 29 × 107 = 46,545
composite factor = 22 × 5 × 23 × 107 = 49,220
composite factor = 24 × 29 × 107 = 49,648
composite factor = 3 × 52 × 23 × 29 = 50,025
composite factor = 24 × 5 × 23 × 29 = 53,360
composite factor = 23 × 3 × 23 × 107 = 59,064
composite factor = 52 × 23 × 107 = 61,525
composite factor = 22 × 5 × 29 × 107 = 62,060
composite factor = 23 × 3 × 52 × 107 = 64,200
composite factor = 22 × 52 × 23 × 29 = 66,700
composite factor = 23 × 29 × 107 = 71,369
composite factor = 2 × 3 × 5 × 23 × 107 = 73,830
composite factor = 23 × 3 × 29 × 107 = 74,472
composite factor = 52 × 29 × 107 = 77,575
composite factor = 23 × 3 × 5 × 23 × 29 = 80,040
composite factor = 2 × 3 × 5 × 29 × 107 = 93,090
composite factor = 23 × 5 × 23 × 107 = 98,440
composite factor = 2 × 3 × 52 × 23 × 29 = 100,050
composite factor = 24 × 3 × 23 × 107 = 118,128
composite factor = 2 × 52 × 23 × 107 = 123,050
composite factor = 23 × 5 × 29 × 107 = 124,120
composite factor = 24 × 3 × 52 × 107 = 128,400
composite factor = 23 × 52 × 23 × 29 = 133,400
composite factor = 2 × 23 × 29 × 107 = 142,738
composite factor = 22 × 3 × 5 × 23 × 107 = 147,660
composite factor = 24 × 3 × 29 × 107 = 148,944
composite factor = 2 × 52 × 29 × 107 = 155,150
composite factor = 24 × 3 × 5 × 23 × 29 = 160,080
composite factor = 3 × 52 × 23 × 107 = 184,575
composite factor = 22 × 3 × 5 × 29 × 107 = 186,180
composite factor = 24 × 5 × 23 × 107 = 196,880
composite factor = 22 × 3 × 52 × 23 × 29 = 200,100
composite factor = 3 × 23 × 29 × 107 = 214,107
composite factor = 3 × 52 × 29 × 107 = 232,725
composite factor = 22 × 52 × 23 × 107 = 246,100
composite factor = 24 × 5 × 29 × 107 = 248,240
composite factor = 24 × 52 × 23 × 29 = 266,800
composite factor = 22 × 23 × 29 × 107 = 285,476
composite factor = 23 × 3 × 5 × 23 × 107 = 295,320
composite factor = 22 × 52 × 29 × 107 = 310,300
composite factor = 5 × 23 × 29 × 107 = 356,845
composite factor = 2 × 3 × 52 × 23 × 107 = 369,150
composite factor = 23 × 3 × 5 × 29 × 107 = 372,360
composite factor = 23 × 3 × 52 × 23 × 29 = 400,200
composite factor = 2 × 3 × 23 × 29 × 107 = 428,214
composite factor = 2 × 3 × 52 × 29 × 107 = 465,450
composite factor = 23 × 52 × 23 × 107 = 492,200
composite factor = 23 × 23 × 29 × 107 = 570,952
composite factor = 24 × 3 × 5 × 23 × 107 = 590,640
composite factor = 23 × 52 × 29 × 107 = 620,600
composite factor = 2 × 5 × 23 × 29 × 107 = 713,690
composite factor = 22 × 3 × 52 × 23 × 107 = 738,300
composite factor = 24 × 3 × 5 × 29 × 107 = 744,720
composite factor = 24 × 3 × 52 × 23 × 29 = 800,400
composite factor = 22 × 3 × 23 × 29 × 107 = 856,428
composite factor = 22 × 3 × 52 × 29 × 107 = 930,900
composite factor = 24 × 52 × 23 × 107 = 984,400
composite factor = 3 × 5 × 23 × 29 × 107 = 1,070,535
composite factor = 24 × 23 × 29 × 107 = 1,141,904
composite factor = 24 × 52 × 29 × 107 = 1,241,200
composite factor = 22 × 5 × 23 × 29 × 107 = 1,427,380
composite factor = 23 × 3 × 52 × 23 × 107 = 1,476,600
composite factor = 23 × 3 × 23 × 29 × 107 = 1,712,856
composite factor = 52 × 23 × 29 × 107 = 1,784,225
composite factor = 23 × 3 × 52 × 29 × 107 = 1,861,800
composite factor = 2 × 3 × 5 × 23 × 29 × 107 = 2,141,070
composite factor = 23 × 5 × 23 × 29 × 107 = 2,854,760
composite factor = 24 × 3 × 52 × 23 × 107 = 2,953,200
composite factor = 24 × 3 × 23 × 29 × 107 = 3,425,712
composite factor = 2 × 52 × 23 × 29 × 107 = 3,568,450
composite factor = 24 × 3 × 52 × 29 × 107 = 3,723,600
composite factor = 22 × 3 × 5 × 23 × 29 × 107 = 4,282,140
composite factor = 3 × 52 × 23 × 29 × 107 = 5,352,675
composite factor = 24 × 5 × 23 × 29 × 107 = 5,709,520
composite factor = 22 × 52 × 23 × 29 × 107 = 7,136,900
composite factor = 23 × 3 × 5 × 23 × 29 × 107 = 8,564,280
composite factor = 2 × 3 × 52 × 23 × 29 × 107 = 10,705,350
composite factor = 23 × 52 × 23 × 29 × 107 = 14,273,800
composite factor = 24 × 3 × 5 × 23 × 29 × 107 = 17,128,560
composite factor = 22 × 3 × 52 × 23 × 29 × 107 = 21,410,700
composite factor = 24 × 52 × 23 × 29 × 107 = 28,547,600
composite factor = 23 × 3 × 52 × 23 × 29 × 107 = 42,821,400
composite factor = 24 × 3 × 52 × 23 × 29 × 107 = 85,642,800
240 factors (divisors)

What times what is 85,642,800?
What number multiplied by what number equals 85,642,800?

All the combinations of any two natural numbers whose product equals 85,642,800.

1 × 85,642,800 = 85,642,800
2 × 42,821,400 = 85,642,800
3 × 28,547,600 = 85,642,800
4 × 21,410,700 = 85,642,800
5 × 17,128,560 = 85,642,800
6 × 14,273,800 = 85,642,800
8 × 10,705,350 = 85,642,800
10 × 8,564,280 = 85,642,800
12 × 7,136,900 = 85,642,800
15 × 5,709,520 = 85,642,800
16 × 5,352,675 = 85,642,800
20 × 4,282,140 = 85,642,800
23 × 3,723,600 = 85,642,800
24 × 3,568,450 = 85,642,800
25 × 3,425,712 = 85,642,800
29 × 2,953,200 = 85,642,800
30 × 2,854,760 = 85,642,800
40 × 2,141,070 = 85,642,800
46 × 1,861,800 = 85,642,800
48 × 1,784,225 = 85,642,800
50 × 1,712,856 = 85,642,800
58 × 1,476,600 = 85,642,800
60 × 1,427,380 = 85,642,800
69 × 1,241,200 = 85,642,800
75 × 1,141,904 = 85,642,800
80 × 1,070,535 = 85,642,800
87 × 984,400 = 85,642,800
92 × 930,900 = 85,642,800
100 × 856,428 = 85,642,800
107 × 800,400 = 85,642,800
115 × 744,720 = 85,642,800
116 × 738,300 = 85,642,800
120 × 713,690 = 85,642,800
138 × 620,600 = 85,642,800
145 × 590,640 = 85,642,800
150 × 570,952 = 85,642,800
174 × 492,200 = 85,642,800
184 × 465,450 = 85,642,800
200 × 428,214 = 85,642,800
214 × 400,200 = 85,642,800
230 × 372,360 = 85,642,800
232 × 369,150 = 85,642,800
240 × 356,845 = 85,642,800
276 × 310,300 = 85,642,800
290 × 295,320 = 85,642,800
300 × 285,476 = 85,642,800
321 × 266,800 = 85,642,800
345 × 248,240 = 85,642,800
348 × 246,100 = 85,642,800
368 × 232,725 = 85,642,800
400 × 214,107 = 85,642,800
428 × 200,100 = 85,642,800
435 × 196,880 = 85,642,800
460 × 186,180 = 85,642,800
464 × 184,575 = 85,642,800
535 × 160,080 = 85,642,800
552 × 155,150 = 85,642,800
575 × 148,944 = 85,642,800
580 × 147,660 = 85,642,800
600 × 142,738 = 85,642,800
642 × 133,400 = 85,642,800
667 × 128,400 = 85,642,800
690 × 124,120 = 85,642,800
696 × 123,050 = 85,642,800
725 × 118,128 = 85,642,800
856 × 100,050 = 85,642,800
870 × 98,440 = 85,642,800
920 × 93,090 = 85,642,800
1,070 × 80,040 = 85,642,800
1,104 × 77,575 = 85,642,800
1,150 × 74,472 = 85,642,800
1,160 × 73,830 = 85,642,800
1,200 × 71,369 = 85,642,800
1,284 × 66,700 = 85,642,800
1,334 × 64,200 = 85,642,800
1,380 × 62,060 = 85,642,800
1,392 × 61,525 = 85,642,800
1,450 × 59,064 = 85,642,800
1,605 × 53,360 = 85,642,800
1,712 × 50,025 = 85,642,800
1,725 × 49,648 = 85,642,800
1,740 × 49,220 = 85,642,800
1,840 × 46,545 = 85,642,800
2,001 × 42,800 = 85,642,800
2,140 × 40,020 = 85,642,800
2,175 × 39,376 = 85,642,800
2,300 × 37,236 = 85,642,800
2,320 × 36,915 = 85,642,800
2,461 × 34,800 = 85,642,800
2,568 × 33,350 = 85,642,800
2,668 × 32,100 = 85,642,800
2,675 × 32,016 = 85,642,800
2,760 × 31,030 = 85,642,800
2,900 × 29,532 = 85,642,800
3,103 × 27,600 = 85,642,800
3,210 × 26,680 = 85,642,800
3,335 × 25,680 = 85,642,800
3,450 × 24,824 = 85,642,800
3,480 × 24,610 = 85,642,800
4,002 × 21,400 = 85,642,800
4,280 × 20,010 = 85,642,800
4,350 × 19,688 = 85,642,800
4,600 × 18,618 = 85,642,800
4,922 × 17,400 = 85,642,800
5,136 × 16,675 = 85,642,800
5,336 × 16,050 = 85,642,800
5,350 × 16,008 = 85,642,800
5,520 × 15,515 = 85,642,800
5,800 × 14,766 = 85,642,800
6,206 × 13,800 = 85,642,800
6,420 × 13,340 = 85,642,800
6,670 × 12,840 = 85,642,800
6,900 × 12,412 = 85,642,800
6,960 × 12,305 = 85,642,800
7,383 × 11,600 = 85,642,800
8,004 × 10,700 = 85,642,800
8,025 × 10,672 = 85,642,800
8,560 × 10,005 = 85,642,800
8,700 × 9,844 = 85,642,800
9,200 × 9,309 = 85,642,800
120 unique multiplications

The final answer:
(scroll down)


85,642,800 has 240 factors (divisors):
1; 2; 3; 4; 5; 6; 8; 10; 12; 15; 16; 20; 23; 24; 25; 29; 30; 40; 46; 48; 50; 58; 60; 69; 75; 80; 87; 92; 100; 107; 115; 116; 120; 138; 145; 150; 174; 184; 200; 214; 230; 232; 240; 276; 290; 300; 321; 345; 348; 368; 400; 428; 435; 460; 464; 535; 552; 575; 580; 600; 642; 667; 690; 696; 725; 856; 870; 920; 1,070; 1,104; 1,150; 1,160; 1,200; 1,284; 1,334; 1,380; 1,392; 1,450; 1,605; 1,712; 1,725; 1,740; 1,840; 2,001; 2,140; 2,175; 2,300; 2,320; 2,461; 2,568; 2,668; 2,675; 2,760; 2,900; 3,103; 3,210; 3,335; 3,450; 3,480; 4,002; 4,280; 4,350; 4,600; 4,922; 5,136; 5,336; 5,350; 5,520; 5,800; 6,206; 6,420; 6,670; 6,900; 6,960; 7,383; 8,004; 8,025; 8,560; 8,700; 9,200; 9,309; 9,844; 10,005; 10,672; 10,700; 11,600; 12,305; 12,412; 12,840; 13,340; 13,800; 14,766; 15,515; 16,008; 16,050; 16,675; 17,400; 18,618; 19,688; 20,010; 21,400; 24,610; 24,824; 25,680; 26,680; 27,600; 29,532; 31,030; 32,016; 32,100; 33,350; 34,800; 36,915; 37,236; 39,376; 40,020; 42,800; 46,545; 49,220; 49,648; 50,025; 53,360; 59,064; 61,525; 62,060; 64,200; 66,700; 71,369; 73,830; 74,472; 77,575; 80,040; 93,090; 98,440; 100,050; 118,128; 123,050; 124,120; 128,400; 133,400; 142,738; 147,660; 148,944; 155,150; 160,080; 184,575; 186,180; 196,880; 200,100; 214,107; 232,725; 246,100; 248,240; 266,800; 285,476; 295,320; 310,300; 356,845; 369,150; 372,360; 400,200; 428,214; 465,450; 492,200; 570,952; 590,640; 620,600; 713,690; 738,300; 744,720; 800,400; 856,428; 930,900; 984,400; 1,070,535; 1,141,904; 1,241,200; 1,427,380; 1,476,600; 1,712,856; 1,784,225; 1,861,800; 2,141,070; 2,854,760; 2,953,200; 3,425,712; 3,568,450; 3,723,600; 4,282,140; 5,352,675; 5,709,520; 7,136,900; 8,564,280; 10,705,350; 14,273,800; 17,128,560; 21,410,700; 28,547,600; 42,821,400 and 85,642,800
out of which 6 prime factors: 2; 3; 5; 23; 29 and 107.
Numbers other than 1 that are not prime factors are composite factors (divisors).
85,642,800 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".