Question

How do you find the prime factorization of the number \[3,240\]?

Answer 1

Hint: In order to find the solution of the given question, that is to the prime factorization of the number \[3,240\]. 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 \[1\].

Complete step by step solution:
According to the question, given number in the question is as follows:
\[3,240\]
The prime factorization of the above number starts by dividing the number by the first prime number. 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 \[3,240\] by \[2\]
\[\Rightarrow 3,240\text{ }\div ~2~=\text{ }1,620~\]- No remainder! \[2\] is one of the factors!
\[\Rightarrow 1,620\text{ }\div ~2~=\text{ }810~~\]- No remainder! \[2\] is one of the factors!
\[\Rightarrow 810\text{ }\div ~2~=\text{ }405~~~\]- No remainder! \[2\] is one of the factors!
\[\Rightarrow 405\text{ }\div \text{ }2\text{ }=\text{ }202.5~~\]- There is a remainder. We can't divide by \[2\] evenly anymore.
Let's try the next prime number
\[\Rightarrow 405\text{ }\div ~3~=\text{ }135~~~\]- No remainder! \[3\] is one of the factors!
\[\Rightarrow 135\text{ }\div ~3~=\text{ }45~~~~\]- No remainder! \[3\] is one of the factors!
\[\Rightarrow 45\text{ }\div ~3~=\text{ }15~~\]- No remainder! \[3\] is one of the factors!
\[\Rightarrow 15\text{ }\div ~3~=\text{ }5~\]- No remainder! \[3\] is one of the factors!
\[\Rightarrow 5\text{ }\div \text{ }3\text{ }=\text{ }1.6667~~\]- There is a remainder. We can't divide by \[3\] evenly anymore. Let's try the next prime number
\[\Rightarrow 5\text{ }\div ~5~=\text{ }1~~\]- No remainder! \[5\] is one of the factors!
We can represent this in the following way also:
\[\begin{align}
  & 2\left| \!{\underline {\,
  3,240 \,}} \right. \\
 & 2\left| \!{\underline {\,
  1,620 \,}} \right. \\
 & 2\left| \!{\underline {\,
  840 \,}} \right. \\
 & 3\left| \!{\underline {\,
  405 \,}} \right. \\
 & 3\left| \!{\underline {\,
  135 \,}} \right. \\
 & 3\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
The divisor(s) on the right side of the symbol \[\div \] above are the prime factors of the number \[3,240\]. If we put all of it together, we have the factors \[2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }5\text{ }=\text{ }3,240\]. It can also be written in exponential form as \[{{2}^{3}}~\times \text{ }{{3}^{4}}~\times \text{ }{{5}^{1}}\].
Therefore, prime factorization of \[3,240\] is \[{{2}^{3}}~\times \text{ }{{3}^{4}}~\times \text{ }{{5}^{1}}\].

Note: Students can make mistakes by making division errors, taking a bigger prime number first and taking an even number or the number which is completely wrong and leads to the wrong answer. It’s important to remember that in prime factorization we divide the number by the first prime number \[2\]. 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 \[1\].