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Greatest Common Factor

Terms

Factor - a factor is any number that can divide into another without leaving a remainder. Since 2 * 3 = 6, 2 and 3 are both factors of 6.
Divisor - Another way of referring to a factor.
Greatest Common Factor - A greatest common factor (or GCF) is the highest factor that is the same between two or more numbers.
Highest Common Factor - Another way of referring to GCF.

Lesson

A Greatest Common Factor (also seen as GCF or HCF) is the highest factor that is the same for two or more numbers. It is easy to figure out the GCF when you split the two numbers up into their factors. For example, if we are told to find the GCF of 15 and 20, we first split 15 and 20 into their factors.

Factors of 15 are 1, 3, 5, 15

Factors of 20 are 1, 2, 4, 5, 10, 20

The factors that appear in both lists are 1 and 5. The greatest of these is 5, so 5 is the Greatest Common Factor of 15 and 20.

Examples

Greatest Common Factor (Example #1)

$20\text{ and }30$
First list the factors of the number (the numbers that divide each number with zero remainder)
Divisors of 20 are: 1, 2, 4, 5, 10, 20
Divisors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30
The largest number that divides $20\text{ and }30$ is $10$, so the GCF = $10$

Greatest Common Factor (Example #2)

$18\text{ and }19$
First list the factors of the number (the numbers that divide each number with zero remainder)
Divisors of 18 are: 1, 2, 3, 6, 9, 18
Divisors of 19 are: 1, 19
The largest number that divides $18\text{ and }19$ is $1$, so the GCF = $1$

Greatest Common Factor (Example #3)

$25\text{ and }50$
First list the factors of the number (the numbers that divide each number with zero remainder)
Divisors of 25 are: 1, 5, 25
Divisors of 50 are: 1, 2, 5, 10, 25, 50
The largest number that divides $25\text{ and }50$ is $25$, so the GCF = $25$

Greatest Common Factor (Example #4)

$10, 15\text{ and }20$
First list the factors of the number (the numbers that divide each number with zero remainder)
Divisors of 10 are: 1, 2, 5, 10
Divisors of 15 are: 1, 3, 5, 15
Divisors of 20 are: 1, 2, 4, 5, 10, 20
The largest number that divides $10, 15\text{ and }20$ is $5$, so the GCF = $5$