Find Greatest Common Divisor of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17, using prime factorization.

Step 1 : Find prime factorization of each number.

$$\begin{aligned}1 =& 1~~~ \color{gray}{\text{ (1 is prime)}} \\[8pt]2 =& 2\\[8pt]3 =& 3\\[8pt]4 =& 2\cdot2\\[8pt]5 =& 5\\[8pt]6 =& 2\cdot3\\[8pt]7 =& 7\\[8pt]8 =& 2\cdot2\cdot2\\[8pt]9 =& 3\cdot3\\[8pt]10 =& 2\cdot5\\[8pt]11 =& 11\\[8pt]12 =& 2\cdot2\cdot3\\[8pt]13 =& 13\\[8pt]14 =& 2\cdot7\\[8pt]15 =& 3\cdot5\\[8pt]16 =& 2\cdot2\cdot2\cdot2\\[8pt]17 =& 17\\[8pt]\end{aligned}$$

(view steps on how to factor 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17. )

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

$$\begin{aligned}1 =& 1\\[8pt]2 =& 2\\[8pt]3 =& 3\\[8pt]4 =& 2\cdot2\\[8pt]5 =& 5\\[8pt]6 =& 2\cdot3\\[8pt]7 =& 7\\[8pt]8 =& 2\cdot2\cdot2\\[8pt]9 =& 3\cdot3\\[8pt]10 =& 2\cdot5\\[8pt]11 =& 11\\[8pt]12 =& 2\cdot2\cdot3\\[8pt]13 =& 13\\[8pt]14 =& 2\cdot7\\[8pt]15 =& 3\cdot5\\[8pt]16 =& 2\cdot2\cdot2\cdot2\\[8pt]17 =& 17\\[8pt]\end{aligned}$$

Note that in this example numbers do not have any common factors.

Step 3 : Multiply the boxed numbers together:

Since there is no boxed numbers we conclude that $~\color{blue}{ \text{GCD = 1} } $.

GCD Calculator