Factors of 321,984. Calculator of Proper, Improper, Prime and Compound Divisors, if Any

All the factors (divisors) of the number 321,984. Connection with the prime factorization of the number

To find all the divisors of the number 321,984:

  • 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 321,984:

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


321,984 = 26 × 32 × 13 × 43
321,984 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 = (6 + 1) × (2 + 1) × (1 + 1) × (1 + 1) = 7 × 3 × 2 × 2 = 84

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

2. Multiply the prime factors of the number 321,984

  • 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
composite factor = 2 × 3 = 6
composite factor = 23 = 8
composite factor = 32 = 9
composite factor = 22 × 3 = 12
prime factor = 13
composite factor = 24 = 16
composite factor = 2 × 32 = 18
composite factor = 23 × 3 = 24
composite factor = 2 × 13 = 26
composite factor = 25 = 32
composite factor = 22 × 32 = 36
composite factor = 3 × 13 = 39
prime factor = 43
composite factor = 24 × 3 = 48
composite factor = 22 × 13 = 52
composite factor = 26 = 64
composite factor = 23 × 32 = 72
composite factor = 2 × 3 × 13 = 78
composite factor = 2 × 43 = 86
composite factor = 25 × 3 = 96
composite factor = 23 × 13 = 104
composite factor = 32 × 13 = 117
composite factor = 3 × 43 = 129
composite factor = 24 × 32 = 144
composite factor = 22 × 3 × 13 = 156
composite factor = 22 × 43 = 172
composite factor = 26 × 3 = 192
composite factor = 24 × 13 = 208
composite factor = 2 × 32 × 13 = 234
composite factor = 2 × 3 × 43 = 258
composite factor = 25 × 32 = 288
composite factor = 23 × 3 × 13 = 312
composite factor = 23 × 43 = 344
composite factor = 32 × 43 = 387
composite factor = 25 × 13 = 416
composite factor = 22 × 32 × 13 = 468
composite factor = 22 × 3 × 43 = 516
composite factor = 13 × 43 = 559
This list continues below...

... This list continues from above
composite factor = 26 × 32 = 576
composite factor = 24 × 3 × 13 = 624
composite factor = 24 × 43 = 688
composite factor = 2 × 32 × 43 = 774
composite factor = 26 × 13 = 832
composite factor = 23 × 32 × 13 = 936
composite factor = 23 × 3 × 43 = 1,032
composite factor = 2 × 13 × 43 = 1,118
composite factor = 25 × 3 × 13 = 1,248
composite factor = 25 × 43 = 1,376
composite factor = 22 × 32 × 43 = 1,548
composite factor = 3 × 13 × 43 = 1,677
composite factor = 24 × 32 × 13 = 1,872
composite factor = 24 × 3 × 43 = 2,064
composite factor = 22 × 13 × 43 = 2,236
composite factor = 26 × 3 × 13 = 2,496
composite factor = 26 × 43 = 2,752
composite factor = 23 × 32 × 43 = 3,096
composite factor = 2 × 3 × 13 × 43 = 3,354
composite factor = 25 × 32 × 13 = 3,744
composite factor = 25 × 3 × 43 = 4,128
composite factor = 23 × 13 × 43 = 4,472
composite factor = 32 × 13 × 43 = 5,031
composite factor = 24 × 32 × 43 = 6,192
composite factor = 22 × 3 × 13 × 43 = 6,708
composite factor = 26 × 32 × 13 = 7,488
composite factor = 26 × 3 × 43 = 8,256
composite factor = 24 × 13 × 43 = 8,944
composite factor = 2 × 32 × 13 × 43 = 10,062
composite factor = 25 × 32 × 43 = 12,384
composite factor = 23 × 3 × 13 × 43 = 13,416
composite factor = 25 × 13 × 43 = 17,888
composite factor = 22 × 32 × 13 × 43 = 20,124
composite factor = 26 × 32 × 43 = 24,768
composite factor = 24 × 3 × 13 × 43 = 26,832
composite factor = 26 × 13 × 43 = 35,776
composite factor = 23 × 32 × 13 × 43 = 40,248
composite factor = 25 × 3 × 13 × 43 = 53,664
composite factor = 24 × 32 × 13 × 43 = 80,496
composite factor = 26 × 3 × 13 × 43 = 107,328
composite factor = 25 × 32 × 13 × 43 = 160,992
composite factor = 26 × 32 × 13 × 43 = 321,984
84 factors (divisors)

What times what is 321,984?
What number multiplied by what number equals 321,984?

All the combinations of any two natural numbers whose product equals 321,984.

1 × 321,984 = 321,984
2 × 160,992 = 321,984
3 × 107,328 = 321,984
4 × 80,496 = 321,984
6 × 53,664 = 321,984
8 × 40,248 = 321,984
9 × 35,776 = 321,984
12 × 26,832 = 321,984
13 × 24,768 = 321,984
16 × 20,124 = 321,984
18 × 17,888 = 321,984
24 × 13,416 = 321,984
26 × 12,384 = 321,984
32 × 10,062 = 321,984
36 × 8,944 = 321,984
39 × 8,256 = 321,984
43 × 7,488 = 321,984
48 × 6,708 = 321,984
52 × 6,192 = 321,984
64 × 5,031 = 321,984
72 × 4,472 = 321,984
78 × 4,128 = 321,984
86 × 3,744 = 321,984
96 × 3,354 = 321,984
104 × 3,096 = 321,984
117 × 2,752 = 321,984
129 × 2,496 = 321,984
144 × 2,236 = 321,984
156 × 2,064 = 321,984
172 × 1,872 = 321,984
192 × 1,677 = 321,984
208 × 1,548 = 321,984
234 × 1,376 = 321,984
258 × 1,248 = 321,984
288 × 1,118 = 321,984
312 × 1,032 = 321,984
344 × 936 = 321,984
387 × 832 = 321,984
416 × 774 = 321,984
468 × 688 = 321,984
516 × 624 = 321,984
559 × 576 = 321,984
42 unique multiplications

The final answer:
(scroll down)


321,984 has 84 factors (divisors):
1; 2; 3; 4; 6; 8; 9; 12; 13; 16; 18; 24; 26; 32; 36; 39; 43; 48; 52; 64; 72; 78; 86; 96; 104; 117; 129; 144; 156; 172; 192; 208; 234; 258; 288; 312; 344; 387; 416; 468; 516; 559; 576; 624; 688; 774; 832; 936; 1,032; 1,118; 1,248; 1,376; 1,548; 1,677; 1,872; 2,064; 2,236; 2,496; 2,752; 3,096; 3,354; 3,744; 4,128; 4,472; 5,031; 6,192; 6,708; 7,488; 8,256; 8,944; 10,062; 12,384; 13,416; 17,888; 20,124; 24,768; 26,832; 35,776; 40,248; 53,664; 80,496; 107,328; 160,992 and 321,984
out of which 4 prime factors: 2; 3; 13 and 43.
Numbers other than 1 that are not prime factors are composite factors (divisors).
321,984 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".