Division of Integers

Recommended Lessons

Not sure where to start? These are the lessons and games we recommend most to students, parents and teachers on Basic-mathematics.com. Each lesson explains the topic in plain language, walks you through worked examples step by step, and ends with practice so you can check your understanding. Pick one below, or use the menu to find lessons on fractions, algebra, geometry, statistics and more.

Division of Integers

In this lesson, you will learn

  • how division is linked to multiplication
  • what the dividend, divisor and quotient are
  • the sign rules for dividing positive and negative integers
  • what happens when zero is part of a division

What Division of Integers Means

Division of integers is the opposite operation of multiplying integers. It is the process by which one is trying to determine how many times a number is contained in another.

Say for instance you do 42 ÷ 6 as shown in the figure below. You are trying to find out how many times 6 is contained in 42.

42 divided by 6 written as long division: 6 is the divisor, 42 is the dividend, and 7, written on top, is the quotient.

Since 6 × 7 = 42, 6 is contained in 42 seven times. Thus, 42 ÷ 6 = 7

Now, it is very important to make the following observations:

quotient × divisor = dividenddividend ÷ divisor = quotientdividend ÷ quotient = divisor

In other words, if the product 4 × 5 = 20, then dividing the product, which is 20, by 4 gives you back 5.

By the same token, dividing the product or 20 by 5 gives you back 4. Having said that, we can use this fact to find the rule for dividing integers.

Use multiplication to find 56 ÷ 8.

Show answer

7. Since 8 × 7 = 56, 8 is contained in 56 seven times.

The Rules for Dividing Integers

Now, let us see how the multiplication facts give us the rules. Since we know that 2 × 6 = 12, then 12 ÷ 6 = 2.

12, 6, and 2 are positive, so we can conclude the following:

positive ÷ positive = positive

Since we know that 2 × −6 = −12, then −12 ÷ −6 = 2.

−12 and −6 are negative, but 2 is positive, so we can conclude the following:

negative ÷ negative = positive

In our previous example, notice also that −12 ÷ 2 = −6

−12 is negative, 2 is positive, and −6 is negative, so we can conclude the following:

negative ÷ positive = negative

Finally, consider −2 × −6 = 12

Then, 12 ÷ −2 = −6

12 is positive, −2 is negative, and −6 is negative, so we can conclude the following:

positive ÷ negative = negative

Notice also that the rule for division of integers is the same as the rule for multiplying integers. Therefore, if you remember the rule for multiplying integers, you already know it for division.

Rule

In a nutshell, here is the most important stuff that you need to remember from this lesson about division of integers:

The division of two integers with the same signs is positive

The division of two integers with different signs is negative

Common mistake

Using the rule for adding integers, where the answer takes the sign of the number farther from zero. In division, only the two signs matter: −20 ÷ 4 = −5, and −20 ÷ −4 = 5, even though 20 is the bigger number in both.

Remember

Zero divided by any non-zero integer is 0, and 0 has no sign: 0 ÷ −7 = 0. Dividing by zero is undefined: −7 ÷ 0 has no answer, because no number times 0 gives −7.

Worked example: Find −72 ÷ −8.
  1. Divide the numbers without their signs: 72 ÷ 8 = 9.
  2. Look at the signs: −72 and −8 are both negative, so the signs are the same.
  3. Same signs give a positive answer, so the quotient is 9.
  4. Check with multiplication: quotient × divisor = 9 × −8 = −72, which is the dividend.

−72 ÷ −8 = 9

What is 45 ÷ −9?

Show answer

−5. 45 ÷ 9 = 5, and the signs are different, so the answer is negative. Check: −5 × −9 = 45.

A Picture Summary of the Rules

Need a little bit more help in order to understand the rule above about division of integers? Here is a summary!

Division of integers. Same signs: 12 ÷ 6 = 2 and −12 ÷ −6 = 2. Different signs: 12 ÷ −6 = −2 and −12 ÷ 6 = −2.

Without working out the answer, is −96 ÷ 12 positive or negative?

Show answer

Negative. −96 is negative and 12 is positive. Different signs give a negative answer (it is −8).

Summary

  • Division is the opposite of multiplication: quotient × divisor = dividend.
  • Same signs give a positive quotient: positive ÷ positive and negative ÷ negative are positive.
  • Different signs give a negative quotient: positive ÷ negative and negative ÷ positive are negative.
  • The sign rules for division are the same as for multiplication.
  • 0 divided by a non-zero integer is 0, and dividing by 0 is undefined.

Test Yourself: 10-Question Quiz

Now check what you learned. Every answer is explained, and each time you play you get new questions.

Question 1 of 10Score: 0

Keep learning