Prime factorization of $$$2178$$$

Find the prime factorization of $$$2178$$$.

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

Determine whether $$$2178$$$ is divisible by $$$2$$$.

It is divisible, thus, divide $$$2178$$$ by $$${\color{green}2}$$$: $$$\frac{2178}{2} = {\color{red}1089}$$$.

Determine whether $$$1089$$$ is divisible by $$$2$$$.

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

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

Determine whether $$$1089$$$ is divisible by $$$3$$$.

It is divisible, thus, divide $$$1089$$$ by $$${\color{green}3}$$$: $$$\frac{1089}{3} = {\color{red}363}$$$.

Determine whether $$$363$$$ is divisible by $$$3$$$.

It is divisible, thus, divide $$$363$$$ by $$${\color{green}3}$$$: $$$\frac{363}{3} = {\color{red}121}$$$.

Determine whether $$$121$$$ is divisible by $$$3$$$.

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

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

Determine whether $$$121$$$ is divisible by $$$5$$$.

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

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

Determine whether $$$121$$$ is divisible by $$$7$$$.

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

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

Determine whether $$$121$$$ is divisible by $$$11$$$.

It is divisible, thus, divide $$$121$$$ by $$${\color{green}11}$$$: $$$\frac{121}{11} = {\color{red}11}$$$.

The prime number $$${\color{green}11}$$$ has no other factors then $$$1$$$ and $$${\color{green}11}$$$: $$$\frac{11}{11} = {\color{red}1}$$$.

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: $$$2178 = 2 \cdot 3^{2} \cdot 11^{2}$$$.

The prime factorization is $$$2178 = 2 \cdot 3^{2} \cdot 11^{2}$$$A.