Common Factors of 420,300 and 980,700. Calculator of Prime and Compound Divisors, if Any

What are the common factors (divisors) of the numbers 420,300 and 980,700?

The common factors of the numbers 420,300 and 980,700 are all the factors of their 'greatest common factor', gcf


Calculate the greatest (highest) common factor (divisor).
Follow the two steps below.

1. Perform the prime factorization of the two numbers:

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


420,300 = 22 × 32 × 52 × 467
420,300 is not a prime number but a composite one.

980,700 = 22 × 3 × 52 × 7 × 467
980,700 is not a prime number but a composite one.


  • Prime number: a natural number that is divisible only by 1 and itself. A prime number has exactly two factors: 1 and 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), ...
  • A composite number is 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)



2. Calculate the greatest (highest) common factor (divisor), gcf, hcf, gcd:

Multiply all the common prime factors, taken by their smallest exponents (the smallest powers).

gcf, hcf, gcd (420,300; 980,700) = 22 × 3 × 52 × 467 = 140,100

» Online calculator. Calculate the greatest (highest) common factor (divisor)



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 = (2 + 1) × (1 + 1) × (2 + 1) × (1 + 1) = 3 × 2 × 3 × 2 = 36

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

3. Multiply the prime factors of the 'gcf':

  • Multiply the prime factors involved in the prime factorization of the GCF in all their unique combinations, that give different results.
  • Also consider the exponents of the prime factors (example: 32 = 3 × 3 = 9).
  • 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 instead.

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 = 2 × 5 = 10
composite factor = 22 × 3 = 12
composite factor = 3 × 5 = 15
composite factor = 22 × 5 = 20
composite factor = 52 = 25
composite factor = 2 × 3 × 5 = 30
composite factor = 2 × 52 = 50
composite factor = 22 × 3 × 5 = 60
composite factor = 3 × 52 = 75
composite factor = 22 × 52 = 100
composite factor = 2 × 3 × 52 = 150
composite factor = 22 × 3 × 52 = 300
This list continues below...

... This list continues from above
prime factor = 467
composite factor = 2 × 467 = 934
composite factor = 3 × 467 = 1,401
composite factor = 22 × 467 = 1,868
composite factor = 5 × 467 = 2,335
composite factor = 2 × 3 × 467 = 2,802
composite factor = 2 × 5 × 467 = 4,670
composite factor = 22 × 3 × 467 = 5,604
composite factor = 3 × 5 × 467 = 7,005
composite factor = 22 × 5 × 467 = 9,340
composite factor = 52 × 467 = 11,675
composite factor = 2 × 3 × 5 × 467 = 14,010
composite factor = 2 × 52 × 467 = 23,350
composite factor = 22 × 3 × 5 × 467 = 28,020
composite factor = 3 × 52 × 467 = 35,025
composite factor = 22 × 52 × 467 = 46,700
composite factor = 2 × 3 × 52 × 467 = 70,050
composite factor = 22 × 3 × 52 × 467 = 140,100
36 common factors (divisors)

What times what is 140,100?
What number multiplied by what number equals 140,100?

All the combinations of any two natural numbers whose product equals 140,100.

1 × 140,100 = 140,100
2 × 70,050 = 140,100
3 × 46,700 = 140,100
4 × 35,025 = 140,100
5 × 28,020 = 140,100
6 × 23,350 = 140,100
10 × 14,010 = 140,100
12 × 11,675 = 140,100
15 × 9,340 = 140,100
20 × 7,005 = 140,100
25 × 5,604 = 140,100
30 × 4,670 = 140,100
50 × 2,802 = 140,100
60 × 2,335 = 140,100
75 × 1,868 = 140,100
100 × 1,401 = 140,100
150 × 934 = 140,100
300 × 467 = 140,100
18 unique multiplications

420,300 and 980,700 have 36 common factors (divisors):
1; 2; 3; 4; 5; 6; 10; 12; 15; 20; 25; 30; 50; 60; 75; 100; 150; 300; 467; 934; 1,401; 1,868; 2,335; 2,802; 4,670; 5,604; 7,005; 9,340; 11,675; 14,010; 23,350; 28,020; 35,025; 46,700; 70,050 and 140,100
out of which 4 prime factors: 2; 3; 5 and 467.
Numbers other than 1 that are not prime factors are composite instead.

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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".