Question

What is the prime factorization of $504$ ?

Answer 1

Hint: Here we are going to find the prime factors of the given number by factoring it. Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.

Complete step-by-step solution:
Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.
Finding the prime factors of $504$,
To find the prime factors, you start by dividing the number by the first prime number, which is$2$. If there is not a remainder, meaning you can divide evenly, then $2$ is a factor of the number.
Continue dividing by $2$ until you cannot divide evenly anymore.
Write down how many $2$'s you were able to divide by evenly. Now try dividing by the next prime factor, which is $3$. The goal is to get to a quotient of $1$.
Here are the first several prime factors: \[2,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}11,{\text{ }}13,{\text{ }}17,{\text{ }}19,{\text{ }}23,{\text{ }}29...\]
Let's start by dividing $504$ by $2$
$504 \div 2 = 252$- No remainder, $2$ is one of the factors
$252 \div 2 = 126$- No remainder, $2$ is one of the factors
$126 \div 2 = 63$ - No remainder, $2$ is one of the factors
$63 \div 2 = 31.5$- There is a remainder. We can't divide by$2$ evenly anymore. Let's try the next prime number
$63 \div 3 = 21$- No remainder, $3$ is one of the factors
$21 \div 3 = 7$- No remainder, $3$ is one of the factors
$7 \div 3 = 2.3333$- There is a remainder. We can't divide by $3$ evenly anymore. Let's try the next prime number
$7 \div 5 = 1.4$- This has a remainder$5$is not a factor.
$7 \div 7 = 1$- No remainder, $7$ is one of the factors
The divisors above are the prime factors of the number \[504\] . If we put all of it together we have the factors \[2 \times 2 \times 2 \times 3 \times 3 \times 7 = 504\] . It can also be written in exponential form as \[{2^3}*{3^{^2}}*{7^1}\].

Note: Another way to do prime factorization is to use a factor tree for \[504\]. That is
$\begin{array}{*{20}{c}}
  {2\left| \!{\underline {\,
  {504} \,}} \right. } \\
  {2\left| \!{\underline {\,
  {252} \,}} \right. } \\
  {2\left| \!{\underline {\,
  {126} \,}} \right. } \\
  {3\left| \!{\underline {\,
  {63} \,}} \right. } \\
  {3\left| \!{\underline {\,
  {21} \,}} \right. } \\
  {7\left| \!{\underline {\,
  7 \,}} \right. } \\
  {\,\,\,\underline 1 }
\end{array}$