Find Greatest Common Divisor of 270 and 75, using prime factorization.
Answer
The GCD of given numbers is 15.
Explanation
Step 1 : Find prime factorization of each number.
$$\begin{aligned}270 =& 2\cdot3\cdot3\cdot3\cdot5\\[8pt]75 =& 3\cdot5\cdot5\\[8pt]\end{aligned}$$(view steps on how to factor 270 and 75. )
Step 2 : Put a box around factors that are common for all numbers:
$$\begin{aligned}270 =& 2\cdot\color{blue}{\boxed{3}}\cdot3\cdot3\cdot\color{red}{\boxed{5}}\\[8pt]75 =& \color{blue}{\boxed{3}}\cdot\color{red}{\boxed{5}}\cdot5\\[8pt]\end{aligned}$$Step 3 : Multiply the boxed numbers together:
$$ GCD = 3\cdot5 = 15 $$This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.
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