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".