## 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 = 2
^{2}× 3 - 60 = 2 × 2 × 3 × 5 = 2
^{2}× 3 × 5 *** Note:**2^{3}= 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. 2^{3}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 = 2
^{2}× 3 - 48 = 2
^{4}× 3 - 360 = 2
^{3}× 3^{2}× 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 = 2
^{3}× 3 - 48 = 2
^{4}× 3 - 360 = 2
^{3}× 3^{2}× 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 = 2
^{2}× 3^{2} - 3,024 = 2
^{4}× 3^{2}× 7 - 5,544 = 2
^{3}× 3^{2}× 7 × 11 - gcf (1,260, 3,024, 5,544) = 2
^{2}× 3^{2}= 252

**And another example:**- 900 = 2
^{2}× 3^{2}× 5^{2} - 270 = 2 × 3
^{3}× 5 - 210 = 2 × 3 × 5 × 7
- gcf (900, 270, 210) = 2 × 3 × 5 = 30

**And one more example:**- 90 = 2 × 3
^{2}× 5 - 27 = 3
^{3} - 22 = 2 × 11
- gcf (90, 27, 22) = 1 - The three numbers have no prime factors in common, they are relatively prime.