GCD calculator

The greatest common division calculator computes GCD of two or more integers using 4 method.

Tutorial

What is Greatest common divisor (GCD)?

The greatest common divisor (factor, multiple) of two integers is the largest number that divides them both.

How to find GCF of two or more numbers?

To find the GCF of two numbers, this calculator uses the following methods:
1. Prime factorization method, 2. Repeated division method, 3. Euclidean algorithm and 4. Listing out the factors.

Examples

Method 1 : Find GCD using prime factorization method

Example: find GCD of 36 and 48

Step 1: find prime factorization of each number:

42 = 2 * 3 * 7

70 = 2 * 5 * 7

Step 2: circle out all common factors:

42 = ② * 3 * ⑦

70 = ② * 5 * ⑦

We see that the GCD is ② * ⑦ = 14

Method 2 : Find GCD using a repeated division

Example: find GCD of 84 and 140.

Step 1: Place the numbers inside division bar:

140

Step 2: Divide both numbers by 2:

2	84	140
	42	70

Step 3: Continue to divide until the numbers do not have a common factor.

②	84	140
②	42	70
⑦	21	35
	3	7

Step 4: The GCD of 84 and 140 is: ② * ② * ⑦ = 28

Method 3 : Euclidean algorithm

Example: Find GCD of 52 and 36, using Euclidean algorithm.

Solution: Divide 52 by 36 and get the remainder, then divide 36 with the remainder from previous step. When the remainder is zero the GCD is the last divisor.

52	:	36	=	1	remainder (16)
		36	:	16	=	1	remainder (4)
				16	:	④	=	4	remainder (0)

We conclude that the GCD = 4.

Method 4 : Listing out the factors

Example: find GCD of 45 and 54 by listing out the factors.

Step 1: Find divisors for the given numbers:

The divisors of 45 are 1, 3, 5, ⑨, 15 and 45

The divisors of 54 are 1, 2, 3, 6, ⑨ 18, 27 and 54

Step 2: The greatest divisor = ⑨

Find Greatest Common Divisor of 170 and 238, using prime factorization.

solution

The GCD of given numbers is 34.

explanation

Step 1 : Find prime factorization of each number.

$$\begin{aligned}170 =& 2\cdot5\cdot17\\[8pt]238 =& 2\cdot7\cdot17\\[8pt]\end{aligned}$$

(view steps on how to factor 170 and 238. )

Step 2 : Put a box around factors that are common for all numbers:

$$\begin{aligned}170 =& \color{blue}{\boxed{2}}\cdot5\cdot\color{red}{\boxed{17}}\\[8pt]238 =& \color{blue}{\boxed{2}}\cdot7\cdot\color{red}{\boxed{17}}\\[8pt]\end{aligned}$$

Step 3 : Multiply the boxed numbers together:

$$ GCD = 2\cdot17 = 34 $$

This solution can be visualized using a Venn diagram.

The GCD equals the product of the numbers at the intersection.