4.1: Greatest Common Divisor

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Think out loud

We want to tile a \(253\) ft by \(123\) ft (\(a=253\) and \(b=123,\) with \(a, b \in \mathbb{Z} \)) floor with identical square tiles. What is the largest square tile we can use?

Definition: Greatest Common Divisor

The greatest common divisor of two integers \(a\) and \(b\), also known as GCD of \(a\) and \(b\), is the greatest positive integer that divides the two integers. We will use the following notation in this class: \(\gcd (a,b)\).

Example \(\PageIndex{1}\):

What is the gcd of \(15\) and \(20\)?

A process to find the solution:
List all positive divisors of \(15\) and \(20\).
The positive divisors of \(15\) are \(1, 3, 5,\) and \(15\).
The positive divisors of \(20\) are \(1, 2, 4, 5, 10,\) and \(20\).
The common positive divisors are \(1, 5\).
As you can see from the list the gcd of \(15\) and \(20\) is \(5.\) That is, the \( \gcd(15, 20)=5.\)

Example \(\PageIndex{2}\):

An elementary gym teacher has \( 3\) grade \(4\) gym classes with \(21, 35\) and \(28\) students in them. The teacher wants to order some equipment that can be used by equal-sized groups in each class. What is the largest group size that will work for all \(3\) classes?

Solution:
You will need to find the GCD of all \(3\) classes.
Firstly, you will find the GCD of \(21\) and \(35.\)
The positive divisors of \(21\) are \(1, 3, 7,\) and \(21\).
The positive divisors of \(35\) are \(1, 5, 7,\) and \(35\).
The GCD of \(21\) and \(35\) is \(7.\)

Since \(7 \mid 28\) you will now find the GCD of \(7\) and \(28\), which turns out to be 7.
Therefore the GCD of \( 21, 35 \) and \(28\) is \(7. \)
Thus, the largest number of students in a group for each class will be \(7.\)

Properties:

Let \(a,b,c \in \mathbb{Z}\). In this section, we have introduced a binary operation on \(\mathbb{Z}\).

Then:

\(\gcd(a, a)=|a|, a\ne 0.\)
\(\gcd(a, b)=\gcd(b, a)\).
\(\gcd(a,b,c)=\gcd (\gcd(a, b), c)=\gcd (a, \gcd(b, c))\).
\(\gcd(a, 0)=a\).
\(\gcd(0, 0)\) is undefined.

In the next section, we will learn an algorithm to calculate GCD.

Practical Uses:

There are many different real-world applications for learning the greatest common divisor (gcd).

Fractions
Algebra
Foundational word problem skills
Ratios
Recipes
Group arrangements