Common Factors of 297,220 and 378,280

The common factors (divisors) of the numbers 297,220 and 378,280?

The common factors of the numbers 297,220 and 378,280 are all the factors of their 'greatest common factor', gcf

Calculations:

Calculate all the divisors (factors) of the given numbers

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

1. Carry out the prime factorization of the two numbers:

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

297,220 = 2² × 5 × 7 × 11 × 193
297,220 is not a prime number but a composite one.

378,280 = 2³ × 5 × 7² × 193
378,280 is not a prime number but a composite one.

» Online calculator. Check whether a number is prime or not. The prime factorization of composite numbers (decomposition into prime factors)

* Prime number: a natural number that is divisible only by 1 and itself. A prime number has exactly two factors: 1 and itself.
* Composite number: a natural number that has at least one other factor than 1 and itself.

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 (297,220; 378,280) = 2² × 5 × 7 × 193 = 27,020

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

How to count the number of factors of a number?

If a number N is prime factorized as:
N = a^m × b^k × c^z
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) × (1 + 1) × (1 + 1) = 3 × 2 × 2 × 2 = 24

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):

neither prime nor composite = 1

prime factor = 2

2² = 4

prime factor = 5

prime factor = 7

2 × 5 = 10

2 × 7 = 14

2² × 5 = 20

2² × 7 = 28

5 × 7 = 35

2 × 5 × 7 = 70

2² × 5 × 7 = 140

This list continues below...

... This list continues from above

prime factor = 193

2 × 193 = 386

2² × 193 = 772

5 × 193 = 965

7 × 193 = 1,351

2 × 5 × 193 = 1,930

2 × 7 × 193 = 2,702

2² × 5 × 193 = 3,860

2² × 7 × 193 = 5,404

5 × 7 × 193 = 6,755

2 × 5 × 7 × 193 = 13,510

2² × 5 × 7 × 193 = 27,020

297,220 and 378,280 have 24 common factors (divisors):
1; 2; 4; 5; 7; 10; 14; 20; 28; 35; 70; 140; 193; 386; 772; 965; 1,351; 1,930; 2,702; 3,860; 5,404; 6,755; 13,510 and 27,020
out of which 4 prime factors: 2; 5; 7 and 193

Other similar operations of calculating common factors:

» What are all the common factors (divisors and prime factors) of the numbers 297,246 and 378,330?

» New: All the Calculations Performed by Our Visitors: Common Divisors of Numbers. Data organized on a Monthly Basis

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".