Prime factorization of $$$3689$$$

Find the prime factorization of $$$3689$$$.

Start with the number $$$2$$$.

Determine whether $$$3689$$$ is divisible by $$$2$$$.

Since it is not divisible, move to the next prime number.

The next prime number is $$$3$$$.

Determine whether $$$3689$$$ is divisible by $$$3$$$.

Since it is not divisible, move to the next prime number.

The next prime number is $$$5$$$.

Determine whether $$$3689$$$ is divisible by $$$5$$$.

Since it is not divisible, move to the next prime number.

The next prime number is $$$7$$$.

Determine whether $$$3689$$$ is divisible by $$$7$$$.

It is divisible, thus, divide $$$3689$$$ by $$${\color{green}7}$$$: $$$\frac{3689}{7} = {\color{red}527}$$$.

Determine whether $$$527$$$ is divisible by $$$7$$$.

Since it is not divisible, move to the next prime number.

The next prime number is $$$11$$$.

Determine whether $$$527$$$ is divisible by $$$11$$$.

Since it is not divisible, move to the next prime number.

The next prime number is $$$13$$$.

Determine whether $$$527$$$ is divisible by $$$13$$$.

Since it is not divisible, move to the next prime number.

The next prime number is $$$17$$$.

Determine whether $$$527$$$ is divisible by $$$17$$$.

It is divisible, thus, divide $$$527$$$ by $$${\color{green}17}$$$: $$$\frac{527}{17} = {\color{red}31}$$$.

The prime number $$${\color{green}31}$$$ has no other factors then $$$1$$$ and $$${\color{green}31}$$$: $$$\frac{31}{31} = {\color{red}1}$$$.

Since we have obtained $$$1$$$, we are done.

Now, just count the number of occurences of the divisors (green numbers), and write down the prime factorization: $$$3689 = 7 \cdot 17 \cdot 31$$$.

The prime factorization is $$$3689 = 7 \cdot 17 \cdot 31$$$A.