Find Greatest Common Divisor of 63, 105 and 210, using prime factorization.

Step 1 : Find prime factorization of each number.

$$\begin{aligned}63 =& 3\cdot3\cdot7\\[8pt]105 =& 3\cdot5\cdot7\\[8pt]210 =& 2\cdot3\cdot5\cdot7\\[8pt]\end{aligned}$$

(view steps on how to factor 63, 105 and 210. )

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

$$\begin{aligned}63 =& \color{blue}{\boxed{3}}\cdot3\cdot\color{red}{\boxed{7}}\\[8pt]105 =& \color{blue}{\boxed{3}}\cdot5\cdot\color{red}{\boxed{7}}\\[8pt]210 =& 2\cdot\color{blue}{\boxed{3}}\cdot5\cdot\color{red}{\boxed{7}}\\[8pt]\end{aligned}$$

Step 3 : Multiply the boxed numbers together:

$$ GCD = 3\cdot7 = 21 $$

This solution can be visualized using a Venn diagram.

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

GCD Calculator