The greatest (highest) common factor (divisor), gcf, hcf, gcd. What it is and how to calculate it.
- Note 1: The greatest common factor (gcf) is also called the highest common factor (hcf), or the greatest common divisor (gcd).
- Note 2: The Prime Factorization of a number: finding the prime numbers that multiply together to make that number.
- Suppose the number "t" evenly divides the number "a" ( = when evenly dividing the number "a" by "t", the remainder is zero).
- When we look at the prime factorization of "a" and "t", we find that:
- 1) all the prime factors of "t" are also prime factors of "a"
- and
- 2) the exponents of the prime factors of "t" are equal to or smaller than the exponents of the prime factors of "a" (see the * Note below)
- For example, the number 12 is a divisor (a factor) of the number 60:
- 12 = 2 × 2 × 3 = 22 × 3
- 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
- * Note: 23 = 2 × 2 × 2 = 8. We say that 2 was raised to the power of 3. In this example, 3 is the exponent and 2 is the base. The exponent indicates how many times the base is multiplied by itself. 23 is the power and 8 is the value of the power.
- If the number "t" is a common divisor of the numbers "a" and "b", then:
- 1) "t" only has the prime factors that also intervene in the prime factorization of "a" and "b".
- and
- 2) each prime factor of "t" has the smallest exponents with respect to the prime factors of the numbers "a" and "b".
- For example, the number 12 is the common divisor of the numbers 48 and 360. Below is their prime factorization:
- 12 = 22 × 3
- 48 = 24 × 3
- 360 = 23 × 32 × 5
- You can see that the number 12 has only the prime factors that also occur in the prime factorization of the numbers 48 and 360.
- You can see above that the numbers 48 and 360 have several common factors: 2, 3, 4, 6, 8, 12, 24. Out of these, 24 is the greatest common factor (GCF) of 48 and 360.
- 24 = 2 × 2 × 2 × 3 = 23 × 3
- 48 = 24 × 3
- 360 = 23 × 32 × 5
- 24, the greatest common factor of the numbers 48 and 360, is calculated as the product of all the common prime factors of the two numbers, taken by the smallest exponents (powers).
- If two numbers "a" and "b" have no other common factor than 1, gcf (a, b) = 1, then the numbers "a" and "b" are called coprime numbers (relatively prime, prime to each other).
- If "a" and "b" are not relatively prime numbers, then every common divisor of "a" and "b" is a divisor of the greatest common divisor of "a" and "b".
- Let's have an example on how to calculate the greatest common factor, gcf, of the following numbers:
- 1,260 = 22 × 32
- 3,024 = 24 × 32 × 7
- 5,544 = 23 × 32 × 7 × 11
- gcf (1,260, 3,024, 5,544) = 22 × 32 = 252
- And another example:
- 900 = 22 × 32 × 52
- 270 = 2 × 33 × 5
- 210 = 2 × 3 × 5 × 7
- gcf (900, 270, 210) = 2 × 3 × 5 = 30
- And one more example:
- 90 = 2 × 32 × 5
- 27 = 33
- 22 = 2 × 11
- gcf (90, 27, 22) = 1 - The three numbers have no prime factors in common, they are relatively prime.