Factors of 312,120. Calculator of Proper, Improper, Prime and Compound Divisors, if Any

All the factors (divisors) of the number 312,120. Connection with the prime factorization of the number

To find all the divisors of the number 312,120:

  • 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 312,120:

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


312,120 = 23 × 33 × 5 × 172
312,120 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) × (3 + 1) × (1 + 1) × (2 + 1) = 4 × 4 × 2 × 3 = 96

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

2. Multiply the prime factors of the number 312,120

  • 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
prime factor = 17
composite factor = 2 × 32 = 18
composite factor = 22 × 5 = 20
composite factor = 23 × 3 = 24
composite factor = 33 = 27
composite factor = 2 × 3 × 5 = 30
composite factor = 2 × 17 = 34
composite factor = 22 × 32 = 36
composite factor = 23 × 5 = 40
composite factor = 32 × 5 = 45
composite factor = 3 × 17 = 51
composite factor = 2 × 33 = 54
composite factor = 22 × 3 × 5 = 60
composite factor = 22 × 17 = 68
composite factor = 23 × 32 = 72
composite factor = 5 × 17 = 85
composite factor = 2 × 32 × 5 = 90
composite factor = 2 × 3 × 17 = 102
composite factor = 22 × 33 = 108
composite factor = 23 × 3 × 5 = 120
composite factor = 33 × 5 = 135
composite factor = 23 × 17 = 136
composite factor = 32 × 17 = 153
composite factor = 2 × 5 × 17 = 170
composite factor = 22 × 32 × 5 = 180
composite factor = 22 × 3 × 17 = 204
composite factor = 23 × 33 = 216
composite factor = 3 × 5 × 17 = 255
composite factor = 2 × 33 × 5 = 270
composite factor = 172 = 289
composite factor = 2 × 32 × 17 = 306
composite factor = 22 × 5 × 17 = 340
composite factor = 23 × 32 × 5 = 360
composite factor = 23 × 3 × 17 = 408
composite factor = 33 × 17 = 459
composite factor = 2 × 3 × 5 × 17 = 510
composite factor = 22 × 33 × 5 = 540
This list continues below...

... This list continues from above
composite factor = 2 × 172 = 578
composite factor = 22 × 32 × 17 = 612
composite factor = 23 × 5 × 17 = 680
composite factor = 32 × 5 × 17 = 765
composite factor = 3 × 172 = 867
composite factor = 2 × 33 × 17 = 918
composite factor = 22 × 3 × 5 × 17 = 1,020
composite factor = 23 × 33 × 5 = 1,080
composite factor = 22 × 172 = 1,156
composite factor = 23 × 32 × 17 = 1,224
composite factor = 5 × 172 = 1,445
composite factor = 2 × 32 × 5 × 17 = 1,530
composite factor = 2 × 3 × 172 = 1,734
composite factor = 22 × 33 × 17 = 1,836
composite factor = 23 × 3 × 5 × 17 = 2,040
composite factor = 33 × 5 × 17 = 2,295
composite factor = 23 × 172 = 2,312
composite factor = 32 × 172 = 2,601
composite factor = 2 × 5 × 172 = 2,890
composite factor = 22 × 32 × 5 × 17 = 3,060
composite factor = 22 × 3 × 172 = 3,468
composite factor = 23 × 33 × 17 = 3,672
composite factor = 3 × 5 × 172 = 4,335
composite factor = 2 × 33 × 5 × 17 = 4,590
composite factor = 2 × 32 × 172 = 5,202
composite factor = 22 × 5 × 172 = 5,780
composite factor = 23 × 32 × 5 × 17 = 6,120
composite factor = 23 × 3 × 172 = 6,936
composite factor = 33 × 172 = 7,803
composite factor = 2 × 3 × 5 × 172 = 8,670
composite factor = 22 × 33 × 5 × 17 = 9,180
composite factor = 22 × 32 × 172 = 10,404
composite factor = 23 × 5 × 172 = 11,560
composite factor = 32 × 5 × 172 = 13,005
composite factor = 2 × 33 × 172 = 15,606
composite factor = 22 × 3 × 5 × 172 = 17,340
composite factor = 23 × 33 × 5 × 17 = 18,360
composite factor = 23 × 32 × 172 = 20,808
composite factor = 2 × 32 × 5 × 172 = 26,010
composite factor = 22 × 33 × 172 = 31,212
composite factor = 23 × 3 × 5 × 172 = 34,680
composite factor = 33 × 5 × 172 = 39,015
composite factor = 22 × 32 × 5 × 172 = 52,020
composite factor = 23 × 33 × 172 = 62,424
composite factor = 2 × 33 × 5 × 172 = 78,030
composite factor = 23 × 32 × 5 × 172 = 104,040
composite factor = 22 × 33 × 5 × 172 = 156,060
composite factor = 23 × 33 × 5 × 172 = 312,120
96 factors (divisors)

What times what is 312,120?
What number multiplied by what number equals 312,120?

All the combinations of any two natural numbers whose product equals 312,120.

1 × 312,120 = 312,120
2 × 156,060 = 312,120
3 × 104,040 = 312,120
4 × 78,030 = 312,120
5 × 62,424 = 312,120
6 × 52,020 = 312,120
8 × 39,015 = 312,120
9 × 34,680 = 312,120
10 × 31,212 = 312,120
12 × 26,010 = 312,120
15 × 20,808 = 312,120
17 × 18,360 = 312,120
18 × 17,340 = 312,120
20 × 15,606 = 312,120
24 × 13,005 = 312,120
27 × 11,560 = 312,120
30 × 10,404 = 312,120
34 × 9,180 = 312,120
36 × 8,670 = 312,120
40 × 7,803 = 312,120
45 × 6,936 = 312,120
51 × 6,120 = 312,120
54 × 5,780 = 312,120
60 × 5,202 = 312,120
68 × 4,590 = 312,120
72 × 4,335 = 312,120
85 × 3,672 = 312,120
90 × 3,468 = 312,120
102 × 3,060 = 312,120
108 × 2,890 = 312,120
120 × 2,601 = 312,120
135 × 2,312 = 312,120
136 × 2,295 = 312,120
153 × 2,040 = 312,120
170 × 1,836 = 312,120
180 × 1,734 = 312,120
204 × 1,530 = 312,120
216 × 1,445 = 312,120
255 × 1,224 = 312,120
270 × 1,156 = 312,120
289 × 1,080 = 312,120
306 × 1,020 = 312,120
340 × 918 = 312,120
360 × 867 = 312,120
408 × 765 = 312,120
459 × 680 = 312,120
510 × 612 = 312,120
540 × 578 = 312,120
48 unique multiplications

The final answer:
(scroll down)


312,120 has 96 factors (divisors):
1; 2; 3; 4; 5; 6; 8; 9; 10; 12; 15; 17; 18; 20; 24; 27; 30; 34; 36; 40; 45; 51; 54; 60; 68; 72; 85; 90; 102; 108; 120; 135; 136; 153; 170; 180; 204; 216; 255; 270; 289; 306; 340; 360; 408; 459; 510; 540; 578; 612; 680; 765; 867; 918; 1,020; 1,080; 1,156; 1,224; 1,445; 1,530; 1,734; 1,836; 2,040; 2,295; 2,312; 2,601; 2,890; 3,060; 3,468; 3,672; 4,335; 4,590; 5,202; 5,780; 6,120; 6,936; 7,803; 8,670; 9,180; 10,404; 11,560; 13,005; 15,606; 17,340; 18,360; 20,808; 26,010; 31,212; 34,680; 39,015; 52,020; 62,424; 78,030; 104,040; 156,060 and 312,120
out of which 4 prime factors: 2; 3; 5 and 17.
Numbers other than 1 that are not prime factors are composite factors (divisors).
312,120 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".