因数

Factors

Factors, also known as divisors, are whole numbers that can be multiplied together to produce another whole number.

Rules about Factors

1. Every integer is a factor of itself
2. $1$ is a factor of any integer
3. Every factor of an integer is less than or equal to the number itself
4. All integers above $1$ has at least $2$ factors: $1$ and the number itself.

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Factor pairs are $2$ factors that will equal to the number itself when multiplied.

The factors of a prime number is $1$ and the number itself, as prime numbers will only have $2$ multiplies.

Let’s take a look at an example.

What are the factors of $12$?

To find the factors of $12$, we first need to think: Which $2$ numbers, when multiplied together, would equal $12$?

Using our rules about factors, we know that all integers above $1$ has at least $2$ factors: $1$ and the number itself. So, a factor pair of $12$ is $1$ and $12$.

Next, we can try the number $2$. $2$ and $6$ can be multiplied together to get the number $12$. Thus, $2$ and $6$ are factors of $12$.

Next, try the number $3$. $3$ and $4$ can be multiplied together to get the number $12$. Therefore, $3$ and $4$ are also factors of $12$.

In conclusion, the factors of $12$ are $1, 2, 3, 4, 6, 12$.

Now, let's try a couple of problems.

To find the factors of $17$, we first need to think: Which $2$ numbers when multiplied together, would equal to $17$?

Using our rules about factors, we know that all integers above $1$ has at least $2$ factors: $1$ and the number itself. So, a factor pair of $17$ is $1$ and $17$.

To find the factors (divisors) of $24$, we first need to think: Which $2$ numbers, when multiplied together, would equal $24$?

Using our rules about factors, we know that all integers above $1$ has at least $2$ factors: $1$ and the number itself. So, a factor pair of $24$ is $1$ and $24$.

Next, we can try the number $2$. $2$ and $12$ can be multiplied together to get the number $24$. Thus, $2$ and $12$ are factors of $24$.

Now, let's try the number $3$. $3$ and $8$ can be multiplied together to get the number $24$. Thus, $3$ and $8$ are factors of $24$.

Next, try the number $4$. $4$ and $6$ equal $24$ when multiplied together, so $4$ and $6$ are factors of $24$.

Therefore, the factors of $24$ are: $1, 2, 3, 4, 6, 8, 12, 24$.

To find the factors (divisors) of $18$, we first need to think: Which $2$ numbers, when multiplied together, would equal $18$?

Using our rules about factors, we know that all integers above $1$ has at least $2$ factors: $1$ and the number itself. So, a factor pair of $18$ is $1$ and $18$.

Next, we can try the number $2$. $2$ and $9$ can be multiplied together to get the number $18$. Thus, $2$ and $9$ are factors of $18$.

Now, let's try the number $3$. $3$ and $6$ can be multiplied together to get the number $18$. Thus, $3$ and $6$ are factors of $18$.

The factors of $18$ are: $1, 2, 3, 6, 9, 18$.

The sum of the factors are: $$\begin{align} 1 + 2 + 3 + 6 + 9 + 18 &= 3+ 3+ 6+ 9+ 18 \\ &= 6 + 6+ 9 +18 \\ &= 12+ 9+ 18 \\ &= 21+ 18 \\ &= 39 \\ \end{align}$$ The sum of the factors of $18$ is $39$.

A factor pair is $2$ factors that when multiplied, will equal the number itself.

We know that $1$ and $30$ are always factors, so $1$ and $30$ are one factor pair.

Next, we can try the number $2$. $2$ and $15$ can be multiplied together to get $30$. Thus, $2$ and $15$ are another factor pair.

Next is the number $3$. $3$ and $10$ can be multiplied together to get $30$. Therefore, $3$ and $10$ are a factor pair.

Next is the number $4$, which cannot be multiplied by another number to equal $30$.

Lastly, we can check the number $5$. $5$ and $6$ can be multiplied together to get $30$, so $5$ and $6$ are a factor pair.

So, the factor pairs for $30$ are: $1$ x $30$, $2$ x $15$, $3$ x $10$, $5$ x $6$. $30$ has $4$ factor pairs.

Divisors

Let's look at the above example a little more closely.

The divisors of $12$ $($also called factors of $12)$ are integers that divide into $12$ without a remainder. $($That means that numbers like $5$ are not divisors of $12,$ since $12 \div 5 = 2$ remainder $2.)$ $$\begin{align} 12 \div 1 &= 12, \text{ so $1$ is a divisor of $12$}\\ 12 \div 2 &= 6, \text{ so $2$ is a divisor of $12$}\\ 12 \div 3 &= 4, \text{ so $3$ is a divisor of $12$}\\ 12 \div 4 &= 3, \text{ so $4$ is a divisor of $12$}\\ 12 \div 6 &= 2, \text{ so $6$ is a divisor of $12$}\\ 12 \div 12 &= 1, \text{ so $12$ is a divisor of $12$}\\ \end{align}$$

$-1, -2, -3, -4, -6,$ and $-12$ are also divisors of $12.$ However, usually you will be dealing with positive divisors.

The positive divisors of $30$ are the positive integers that divide into $30$ without a remainder. $$\begin{align} 30 \div 1 &= 30, \text{ so $1$ is a divisor of $30$}\\ 30 \div 2 &= 15, \text{ so $2$ is a divisor of $30$}\\ 30 \div 3 &= 10, \text{ so $3$ is a divisor of $30$}\\ 30 \div 5 &= 6, \text{ so $5$ is a divisor of $30$}\\ 30 \div 6 &= 5, \text{ so $6$ is a divisor of $30$}\\ 30 \div 10 &= 3, \text{ so $10$ is a divisor of $30$}\\ 30 \div 15 &= 2, \text{ so $15$ is a divisor of $30$}\\ 30 \div 30 &= 1, \text{ so $30$ is a divisor of $30$}\\ \end{align}$$

Numbers like $4$ are not divisors of $30,$ since $30 \div 4 = 7$ remainder $2.$

$0$ is not a divisor of $30$ since $30 \div 0$ yields an undefined answer. In fact, $0$ is not a divisor of any number, since any number divided by $0$ yields an undefined answer.

An important note is that an integer always has divisors of $1$ and itself (besides $0,$ which does not have a divisor of $0).$

Bonus: Factors aren't just for integers! You can also factor monomials!

(Video) Factoring

by Math Antics

This video by mathantics explains the definition of a factor and how to find them.

$\large{\text{Summary}}$

Factoring is like being given the answer to a multiplication problem. The goal is to figure out what that multiplication problem was. We can think of factoring as unmultiplying. We take one number and find what two numbers we can multiply together to make it.

Suppose we want to factor the number $10$. We can look at this as asking, what two numbers multiply together to make $10$?

Well $2\times5$ equals $10$. So we say that $2$ and $5$ are factors of $10$.

But wait, there's still some more. If we multiply $1\times10$ this also equals $10$. So $1$ and $10$ are factors of $10$.

This leads us to an important rule. A whole number always has at least two factors: $1$ and itself.

Divisors

Factors are also known as divisors. This means they can divide evenly into the number they're a factor of. In other words, when you divide by them, they don't leave a remainder.

For example, $4$ is a factor of the number $36$. If we divide $4$ into $36$, we get $9$ and no remainder. This also means that $9$ is a factor of $36$.

If we divide $5$ into $36$, we get $7$ with a remainder of $1$. Therefore, $5$ is not a factor.