6: Calculate all the factors (divisors) of the number (proper, improper and the prime factors)
The factors (divisors) of the number 6
6 is a composite number and can be prime factorized. So what are all the factors (divisors) of the number 6?
A factor (a divisor) of the number 6 is a natural number B which when multiplied by another natural number C equals the given number 6:
6 = B × C. Example: 60 = 2 × 30.
Both B and C are factors of 6.
To find all the factors (divisors) of the number 6:
1) Break down the number into its prime factors (build the number's prime factorization, decompose it into prime factors, write it as a product of prime numbers).
2) Then multiply these prime factors in all their unique combinations, that yield different results.
1) The prime factorization:
The prime factorization of the number 6 (the decomposition of the number into prime factors, breaking the number down into prime numbers) = dividing the number 6 into smaller, prime numbers. The number 6 results from the multiplication of these prime numbers.
6 = 2 × 3
6 is not a prime number but a composite one.
* The natural numbers that are divisible (that are divided evenly) only by 1 and themselves are called prime numbers. Examples: 2, 3, 5, 7, 11, 13, 17. A prime number has exactly two factors: 1 and the number itself.
* A composite number is a natural number that has at least one factor other than 1 and itself. Examples: 4, 6, 8, 9, 10, 12, 14.
2) How do I find all the factors (divisors) of the number?
Multiply the prime factors involved in the prime factorization of the number in all their unique combinations, that give different results.
6 = 2 × 3
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
prime factor =
3
2 × 3 =
6
The final answer:
(scroll down)
6 has 4 factors (divisors):
1; 2; 3 and 6
out of which 2 prime factors: 2 and 3
6 and 1 are called by some authors improper factors (improper divisors), 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.
Other similar operations of calculating factors (divisors):
The latest 5 sets of calculated factors (divisors): of one number or the common factors of two numbers
Calculate all the divisors (factors) of the given numbers
How to calculate (find) all the factors (divisors) of a number:
Break down the number into prime factors. Then multiply its prime factors in all their unique combinations, that give different results.
To calculate the common factors of two numbers:
The common factors (divisors) of two numbers are all the factors of the greatest common factor, gcf.
Calculate the greatest (highest) common factor (divisor) of the two numbers, gcf (hcf, gcd).
Break down the GCF into prime factors. Then multiply its prime factors in all their unique combinations, that give different results.
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".
Some articles on the prime numbers