质因数分解

(Video) Prime Factorization

by mathman1024

This video by mathman1024 covers prime factorization.

$\large{\text{Summary}}$

A prime is a number with only two distinct factors: $1$ and itself. In other words, it only has two numbers that can divide into it evenly which are $1$ and itself.

A composite is a number with more than two distinct factors.

Let's list the factors of the number $12$. These are the numbers that can divide evenly into $12$. $$1,2,3,4,6,12$$ The number $12$ has more than two distinct factors, therefore we can say it's a composite number.

Let's look at the number $7$ now. $$1,7$$ The only numbers that go into $7$ evenly are $1$ and itself. Therefore, $7$ is a prime number.

The number $1$ is a special case. It is neither prime nor composite.

Prime factorization is when you break a number down into its prime factors.

Let's find the prime factorization of the number $12$. One way to do these is to make a factor tree. We start by putting $12$ at the top and creating two branches for two factors of the number. For the number $12$ we can choose $3$ and $4$ or $2$ and $6$. Let's choose $2$ and $6$.

Now, we continue branching until all the numbers are primes. Since $2$ is a prime number, we're done with that branch. $6$, on the other hand, is not a prime, so we have to factor that one.

Both branches of the $6$ are prime numbers so we are done with our prime factorization. You just take the digits at the end of each branch and they are your answer. The prime factorization of $12$ is: $$\begin{align} 12&=2\cdot2\cdot3\\ 12&=2^2\cdot3 \end{align}$$ You do not have to write it using exponents but it is usually a good idea to do it, especially when the numbers get larger.

Try finding the prime factorization of this next number on your own. Then, watch the video to see the answer!

1. $54$

As you may already know, a number can have a lot of factors. A bunch of factors multiplied together to represent a number is called factorization.

What is Prime Factorization?

Well prime factorization just a unique way of showing the factorization of a number-by reducing the number into a series of only prime numbers multiplied together.

Let's take the number $60$. A random factorization of $60$ could be $2\times3\times10$ or $5\times4\times3$. However the prime factorization of $60$ would be $2\times2\times3\times5$ (Note how $2$, $3$, and $5$ are all prime).

By convention, the prime factors are written in numerical order and common factors are expressed as exponents.

Finding Prime Factors

One way is to divide the number by prime factors until you are left with a prime number. Personally, I start dividing by the lowest prime factor each time and work my way up.

Instead of me explaining through text all the time, let us do one example. We have the number $60$. To find the prime factorization of $60$ we start off by looking at the smallest prime $2$.

Is $60$ divisible by $2$? Yes. So $2$ is a prime factor of $60$. Then we divide $60$ by $2$.

Now we have $30$. Again we have to check $2$ because a prime factor can be counted more than once.

Is $30$ divisible by $2$? Yes. So $2$ is a prime factor of $60$. Then we divide $30$ by $2$.

Now we have $15$. After checking $2$ again, we find that $2$ is no longer a factor so we go onto the next prime $3$.

Is $15$ divisible by $3$? Yes. So $3$ is a prime factor of $60$. Then we divide $15$ by $3$.

Now we have $5$. Since $5$ is already a prime number we are done.

Multiplying the prime factors we obtained previously, we have $2\times2\times3\times5=2^2\times3\times5$.

Before we begin with prime factorization let's review the concepts of prime numbers and factoring.

Factoring is a way to break a number down by finding smaller numbers that can be multiplied together to get that number. i.e. $3$ and $12$ can be multiplied together to get $36$ so they are factors of $36$.

Prime Numbers are numbers that only have two factors, one and itself i.e. the only factors of $7$ are $1$ and $7$ so seven is prime.

When we are asked to find the prime factorization of a number we are being asked to break a composite number down into only prime numbers. This is done by finding a factor pair of the given composite number and then factoring that factor pair until we are left with only prime numbers.

I like to use a visual called a factor tree to help me find all of the prime factors of a number. Below, I used a factor tree to find the prime factorization of $36$.

At the top of the tree is the number we are given, in this case $36$. Next, we find a factor pair. I chose to use $6$ and $6$. Since six is composite, we must find the factors of six. The next factor pair is $2$ and $3$. Since both $2$ and $3$ are prime, we can not factor anymore. Thus, the prime factors of $36$ are $2,2,3,3$.

We can check if we have the correct prime factorization by multiplying all of the prime factors together; if the product is the original number then you are done and you found all of the prime factors. Indeed, in our example, $2 \times 3 \times 2 \times 3 = 2^2 \times 3^2 = 36.$

Bonus: You can also use factor trees to factor monomials!