Factors of 49,680. Calculator of Proper, Improper, Prime and Compound Divisors, if Any

All the factors (divisors) of the number 49,680. Connection with the prime factorization of the number

To find all the divisors of the number 49,680:

  • 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 49,680:

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


49,680 = 24 × 33 × 5 × 23
49,680 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 = (4 + 1) × (3 + 1) × (1 + 1) × (1 + 1) = 5 × 4 × 2 × 2 = 80

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

2. Multiply the prime factors of the number 49,680

  • 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
composite factor = 22 × 3 = 12
composite factor = 3 × 5 = 15
composite factor = 24 = 16
composite factor = 2 × 32 = 18
composite factor = 22 × 5 = 20
prime factor = 23
composite factor = 23 × 3 = 24
composite factor = 33 = 27
composite factor = 2 × 3 × 5 = 30
composite factor = 22 × 32 = 36
composite factor = 23 × 5 = 40
composite factor = 32 × 5 = 45
composite factor = 2 × 23 = 46
composite factor = 24 × 3 = 48
composite factor = 2 × 33 = 54
composite factor = 22 × 3 × 5 = 60
composite factor = 3 × 23 = 69
composite factor = 23 × 32 = 72
composite factor = 24 × 5 = 80
composite factor = 2 × 32 × 5 = 90
composite factor = 22 × 23 = 92
composite factor = 22 × 33 = 108
composite factor = 5 × 23 = 115
composite factor = 23 × 3 × 5 = 120
composite factor = 33 × 5 = 135
composite factor = 2 × 3 × 23 = 138
composite factor = 24 × 32 = 144
composite factor = 22 × 32 × 5 = 180
composite factor = 23 × 23 = 184
composite factor = 32 × 23 = 207
composite factor = 23 × 33 = 216
This list continues below...

... This list continues from above
composite factor = 2 × 5 × 23 = 230
composite factor = 24 × 3 × 5 = 240
composite factor = 2 × 33 × 5 = 270
composite factor = 22 × 3 × 23 = 276
composite factor = 3 × 5 × 23 = 345
composite factor = 23 × 32 × 5 = 360
composite factor = 24 × 23 = 368
composite factor = 2 × 32 × 23 = 414
composite factor = 24 × 33 = 432
composite factor = 22 × 5 × 23 = 460
composite factor = 22 × 33 × 5 = 540
composite factor = 23 × 3 × 23 = 552
composite factor = 33 × 23 = 621
composite factor = 2 × 3 × 5 × 23 = 690
composite factor = 24 × 32 × 5 = 720
composite factor = 22 × 32 × 23 = 828
composite factor = 23 × 5 × 23 = 920
composite factor = 32 × 5 × 23 = 1,035
composite factor = 23 × 33 × 5 = 1,080
composite factor = 24 × 3 × 23 = 1,104
composite factor = 2 × 33 × 23 = 1,242
composite factor = 22 × 3 × 5 × 23 = 1,380
composite factor = 23 × 32 × 23 = 1,656
composite factor = 24 × 5 × 23 = 1,840
composite factor = 2 × 32 × 5 × 23 = 2,070
composite factor = 24 × 33 × 5 = 2,160
composite factor = 22 × 33 × 23 = 2,484
composite factor = 23 × 3 × 5 × 23 = 2,760
composite factor = 33 × 5 × 23 = 3,105
composite factor = 24 × 32 × 23 = 3,312
composite factor = 22 × 32 × 5 × 23 = 4,140
composite factor = 23 × 33 × 23 = 4,968
composite factor = 24 × 3 × 5 × 23 = 5,520
composite factor = 2 × 33 × 5 × 23 = 6,210
composite factor = 23 × 32 × 5 × 23 = 8,280
composite factor = 24 × 33 × 23 = 9,936
composite factor = 22 × 33 × 5 × 23 = 12,420
composite factor = 24 × 32 × 5 × 23 = 16,560
composite factor = 23 × 33 × 5 × 23 = 24,840
composite factor = 24 × 33 × 5 × 23 = 49,680
80 factors (divisors)

What times what is 49,680?
What number multiplied by what number equals 49,680?

All the combinations of any two natural numbers whose product equals 49,680.

1 × 49,680 = 49,680
2 × 24,840 = 49,680
3 × 16,560 = 49,680
4 × 12,420 = 49,680
5 × 9,936 = 49,680
6 × 8,280 = 49,680
8 × 6,210 = 49,680
9 × 5,520 = 49,680
10 × 4,968 = 49,680
12 × 4,140 = 49,680
15 × 3,312 = 49,680
16 × 3,105 = 49,680
18 × 2,760 = 49,680
20 × 2,484 = 49,680
23 × 2,160 = 49,680
24 × 2,070 = 49,680
27 × 1,840 = 49,680
30 × 1,656 = 49,680
36 × 1,380 = 49,680
40 × 1,242 = 49,680
45 × 1,104 = 49,680
46 × 1,080 = 49,680
48 × 1,035 = 49,680
54 × 920 = 49,680
60 × 828 = 49,680
69 × 720 = 49,680
72 × 690 = 49,680
80 × 621 = 49,680
90 × 552 = 49,680
92 × 540 = 49,680
108 × 460 = 49,680
115 × 432 = 49,680
120 × 414 = 49,680
135 × 368 = 49,680
138 × 360 = 49,680
144 × 345 = 49,680
180 × 276 = 49,680
184 × 270 = 49,680
207 × 240 = 49,680
216 × 230 = 49,680
40 unique multiplications

The final answer:
(scroll down)


49,680 has 80 factors (divisors):
1; 2; 3; 4; 5; 6; 8; 9; 10; 12; 15; 16; 18; 20; 23; 24; 27; 30; 36; 40; 45; 46; 48; 54; 60; 69; 72; 80; 90; 92; 108; 115; 120; 135; 138; 144; 180; 184; 207; 216; 230; 240; 270; 276; 345; 360; 368; 414; 432; 460; 540; 552; 621; 690; 720; 828; 920; 1,035; 1,080; 1,104; 1,242; 1,380; 1,656; 1,840; 2,070; 2,160; 2,484; 2,760; 3,105; 3,312; 4,140; 4,968; 5,520; 6,210; 8,280; 9,936; 12,420; 16,560; 24,840 and 49,680
out of which 4 prime factors: 2; 3; 5 and 23.
Numbers other than 1 that are not prime factors are composite factors (divisors).
49,680 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".