Positive Integer Division — Rules & Examples

Intuition

Division formalizes the notion of evenly distributing or splitting some number of objects into some number of groups.

Suppose we have $6$ cowboy hats and $2$ cowboys and we want to give both cowboys the same number of hats. We can divide the number of cowboy hats by the number of cowboys to get that each cowboy gets $3$ hats. Mathematically, this is: $$6\div 2=3$$ We can also write: $$6\div 3=2$$ This represents the idea that $6$ cowboy hats, when divided into groups of $3,$ yields $2$ such groups.

Opposite of multiplication

Division can also be seen as the reverse of multiplication. In the previous cowboy hat example, we know that there are $2$ cowboys with $3$ cowboy hats each. This gives $6$ cowboy hats among the two of them.

This example tells us that the following equations tell us equivalent information: $$6\div 2=3$$ $$6\div 3=2$$ $$2\times 3=6$$ $$3\times 2=6$$ In each situation, there are $6$ hats, $2$ cowboys, and $3$ hats per cowboy.

Anatomy of division

Division consists of three elements: the dividend, the divisor, and the quotient. The dividend is the total number that gets divided by the divisor, which results in the quotient. There are several ways to denote the division operation:

$$\text{dividend}\div\text{divisor}=\text{quotient}$$ $$\text{dividend}/\text{divisor}=\text{quotient}$$ $$\frac{\text{dividend}}{\text{divisor}}=\text{quotient}$$

The division as the reverse of multiplication framework tells us that the product of the quotient and the divisor gives the dividend.

Dividing with remainders

So what happens if the divisor does not evenly divide the dividend? One way to approach this issue is to use remainders - this process involves making the quotient as large as possible and seeing what remains.

Let's say we have those two cowboys again, but now there are $7$ cowboy hats. We can evenly give each cowboy $3$ hats, but then there's $7-2\times 3=1$ extra hat left over. We represent this information with the following expression: $$7\div 2=3 \text{ R } 1$$ This tells us that there are $7$ hats total, $2$ cowboys total, $3$ cowboy hats per cowboy, and $1$ hat remaining after the hats are distributed. If we put the remainder over the $2$, we get the fractional part. In this case, the fractional part is $\frac{1}{2}$.

Here's a graphic with another example:

(Video) Dividing Integers

by mathman1024

Here is a video by mathman1024 that goes over dividing positive integers, dividing negative integers, and dividing a negative and a positive integer.

$\large{\text{Summary}}$

He starts by going over some of the basics of division. Division has $3$ parts: the divisor, the dividend, and the quotient.

Division will normally be written one of two ways: $$\begin{align} 15\div3&=5\\ \\ {\text{dividend}\rightarrow\atop \text{divisor}\rightarrow }\frac{15}{3}&=5\leftarrow \text{quotient} \end{align}$$

Division follows the sign rules given by multiplication. $$\begin{align} \frac{+}{+} &= +\\ \\ \frac{-}{-}&=+\\ \\ \frac{+}{-}&=-\\ \\ \frac{-}{+}&=-\end{align}$$ Dividing two numbers with the same sign will give a positive answer, while dividing two numbers with different signs will give a negative answer.

This video doesn't give a specific example dividing two positive numbers, so let's try one on our own.

Dividing is the opposite of multiplying. In a division problem, the answer is how many times you can add a number into another (rather than what you get when you add a number a bunch of times).

For smaller numbers, you can do this by just seeing how many times a number can fit into another. For bigger ones, you'll have to use different strategies, but right now we're going to focus on the basic concepts.

It'll be easier to understand division with some examples. Let's work on the problem $8 \div 2$.

Perfect Division

Just like with subtraction, the first number in the expression is the one we're going to adjust. So, let's start by finding $8$ on the number line.

Now, we want to see how many times $2$ can fit into $8$. Let's find $2$ on the number line:

We can see that $2$ is less than $8$, and that we might be able to add it to itself a couple times before we get to $8$. So let's do that, and count how many times we can add $2$ before we fill—or exceed—$8$.

We can add $2$ to itself $4$ times before it's equal to $8$. So, $8 \div 2 = 4$.

Remainders

Sometimes, though, the number you're dividing by won't fit perfectly into the number you're dividing. When that happens, you'll end up with a remainder. A remainder is the amount that's left over after you take as much as you can out of the number you're dividing. So, for example, if I were solving the problem $7 \div 3$, I could add $3$ into $7$ twice, but not a third time. But $3 \times 2 = 6$, so $1$ would be the remainder.

In other words, $7 \div 3 = 2$ with a remainder of $1$.

Let's look at that on the number line.

As before, the number we're dividing is $7$, so we have to find that first.

Next, let's find $3$.

Now, we can see that $3$ is less than $7$. Let's see how many times we can add it in.

We can add $3$ into $7$ fully twice, but then it overflows during the third addition. We can see, though, that adding $3$ twice only gets us to $6$, and there's still $1$ step to go to reach $7$. Because of this, we say that $7 \div 3 = 2$ with remainder $1$.