Greatest Common Factors

Hi. This page will be about greatest common factors between numbers (and variables).


Table Of Contents


What Is A Factor?

A factor is a positive whole number which (exactly) divides into a whole number with a zero remainder. In other words, a whole number divided by a factor gives another whole number exactly. As an example, any even number can be divided by 2 with zero remainder.

Here is an example. The number 4 has factors of 1, 2 and 4 itself. These factors divide into 4 exactly since 4 \div 1 = 4, 4 \div 2 = 2, and 4 \div 4 = 1.


Greatest Common Factors

Suppose we have two whole numbers (integers) a and b. The greatest common factor (GCF) is the largest factor in which the numbers a and b share. The GCF can also be seen as the largest number which divides into both numbers a and b with zero remainder.

As an example, the numbers 10 and 20 will be used. The factors of 10 are 1, 2, 5, and 10 itself while the factors of 20 are 1, 2, 4, 5, 10 and 20 itself. The largest factor shared between 10 and 20 is 10. (20 is the largest factor for 20 but it does not exist for 10.) The GCF of 10 and 20 would be 10.

We can extend this to three numbers. Suppose we have three numbers 4, 10 and 22. The factors of 4 are 1, 2 and 4, the factors of 10 are 1, 2, 5, and 10, and the factors of 22 are 1, 2, 11 and 22. In this case, the GCF of 4, 10 and 22 is 2.

Greatest common factors applies to variables too. Factors of x^3 are 1, x, x^2, x^3 and factors of x^2 are 1, x, and x^2. The GCF of x^3 and x^2 is x^2.


Relatively Prime Numbers

The concept of relatively prime (mutally prime, coprime or co-prime) is not a typical high school mathematics topic. (I was not taught this in my high school.) However, this concept is not that difficult. Do not confuse relatively prime numbers with prime numbers. Here is a brief overview of relatively prime numbers.

Two numbers a and b are relatively prime if their greatest common factor is 1.

Examples of relatively prime numbers include 2 and 3, 14 and 17.


Practice Problems

Here are some practice problems with answers in the next section. Questions 5, 6 and 7 are a bit more theoretical in nature.

1) What is the greatest common factor of 16 and 32?

2) What is the greatest common factor of 3 and 9?

3) What is the greatest common factor of x and \sqrt{x}?

4)What is the greatest common factor of 8x^2, 4x^2 and 2x?

5) Are 7 and 100 relatively prime numbers?

6) Give two pairs of relatively prime numbers which are least 5 but less than 30.

7) Suppose you are given two prime numbers where a prime number is a whole number (at least 1) which has 2 factors 1 and itself. Are any two prime numbers relatively prime? If not, give an example where two prime numbers are not relatively prime.


Answers

1) Factors of 16 are 1, 2, 4, 8 and 16 and factors of 32 are 1, 2, 4, 8, 16 and 32. The GCF is 16.

2) 3

3) Remember that \sqrt{x} = x^{1/2}. The GCF is \sqrt{x}.

4) We consider both the number and the variable x here. The GCF here is 2x.

5) The factors of 7 are 1 and 7 and the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100. The greatest common factor between 7 and 10 is 1. Yes, 7 and 10 are relatively prime numbers

6) Examples include 5 and 6, 10 and 21 and 11 and 27.

7) The answer here is yes. A prime number has two factors which are 1 and itself. Two prime numbers are relatively prime if 1 is the greatest common factor. If prime number a had factors of 1, itself (a) and prime number b then b would be the GCF but a would no longer be a prime number with that third factor.