Find Greatest Common Divisor of 900, 216, 36, 120, 144, 108, 49, 60, 150 and 25, using prime factorization.

Step 1 : Find prime factorization of each number.

$$\begin{aligned}900 =& 2\cdot2\cdot3\cdot3\cdot5\cdot5\\[8pt]216 =& 2\cdot2\cdot2\cdot3\cdot3\cdot3\\[8pt]36 =& 2\cdot2\cdot3\cdot3\\[8pt]120 =& 2\cdot2\cdot2\cdot3\cdot5\\[8pt]144 =& 2\cdot2\cdot2\cdot2\cdot3\cdot3\\[8pt]108 =& 2\cdot2\cdot3\cdot3\cdot3\\[8pt]49 =& 7\cdot7\\[8pt]60 =& 2\cdot2\cdot3\cdot5\\[8pt]150 =& 2\cdot3\cdot5\cdot5\\[8pt]25 =& 5\cdot5\\[8pt]\end{aligned}$$

(view steps on how to factor 900, 216, 36, 120, 144, 108, 49, 60, 150 and 25. )

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

$$\begin{aligned}900 =& 2\cdot2\cdot3\cdot3\cdot5\cdot5\\[8pt]216 =& 2\cdot2\cdot2\cdot3\cdot3\cdot3\\[8pt]36 =& 2\cdot2\cdot3\cdot3\\[8pt]120 =& 2\cdot2\cdot2\cdot3\cdot5\\[8pt]144 =& 2\cdot2\cdot2\cdot2\cdot3\cdot3\\[8pt]108 =& 2\cdot2\cdot3\cdot3\cdot3\\[8pt]49 =& 7\cdot7\\[8pt]60 =& 2\cdot2\cdot3\cdot5\\[8pt]150 =& 2\cdot3\cdot5\cdot5\\[8pt]25 =& 5\cdot5\\[8pt]\end{aligned}$$

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} } $.

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