We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know 12÷4=312÷4=3 because 3·4=12.3·4=12. Knowing all the multiplication number facts is very important when doing division.
We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. In Example 2, we know 24÷8=324÷8=3 is correct because 3·8=24.3·8=24.
Divide. Then check by multiplying. ⓐ 42÷642÷6 ⓑ 729729 ⓒ 763763

ⓐ  

42÷642÷6 
Divide 42 by 6. 
77 
Check by multiplying.
7·67·6 

42✓42✓ 


ⓑ  

729729 
Divide 72 by 9. 
88 
Check by multiplying. 8·98·9 

72✓72✓ 


ⓒ  

763763 
Divide 63 by 7. 
99 
Check by multiplying.
9·79·7 

63✓63✓ 

Divide. Then check by multiplying:
ⓐ 54÷654÷6 ⓑ 279279
Divide. Then check by multiplying:
ⓐ 369369 ⓑ 840840
What is the quotient when you divide a number by itself?
1515=1because1·15=151515=1because1·15=15
Dividing any number (except 0)(except 0) by itself produces a quotient of 1.1. Also, any number divided by 11 produces a quotient of the number. These two ideas are stated in the Division Properties of One.
Any number (except 0) divided by itself is one. 
a÷a=1a÷a=1 
Any number divided by one is the same number. 
a÷1=aa÷1=a 
Divide. Then check by multiplying:
 ⓐ 11÷1111÷11
 ⓑ 191191
 ⓒ 1717

ⓐ  

11÷1111÷11 
A number divided by itself is 1. 
11 
Check by multiplying. 1·111·11 

11✓11✓ 


ⓑ  

191191 
A number divided by 1 equals itself. 
1919 
Check by multiplying. 19·119·1 

19✓19✓ 


ⓒ  

1717 
A number divided by 1 equals itself. 
77 
Check by multiplying. 7·17·1 

7✓7✓ 

Divide. Then check by multiplying:
ⓐ 14÷1414÷14 ⓑ 271271
Divide. Then check by multiplying:
ⓐ 161161 ⓑ 1414
Suppose we have $0,$0, and want to divide it among 33 people. How much would each person get? Each person would get $0.$0. Zero divided by any number is 0.0.
Now suppose that we want to divide $10$10 by 0.0. That means we would want to find a number that we multiply by 00 to get 10.10. This cannot happen because 00 times any number is 0.0. Division by zero is said to be undefined.
These two ideas make up the Division Properties of Zero.
Zero divided by any number is 0. 
0÷a=00÷a=0 
Dividing a number by zero is undefined. 
a÷0a÷0 undefined 
Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away 00 from 10?10? Because subtracting 00 will never change the total, we will never get an answer. So we cannot divide a number by 0.0.
Divide. Check by multiplying: ⓐ 0÷30÷3 ⓑ 10/0.10/0.

ⓐ  

0÷30÷3 
Zero divided by any number is zero. 
00 
Check by multiplying. 0·30·3 

0✓0✓ 


ⓑ  

10/010/0 
Division by zero is undefined. 
undefined 
Divide. Then check by multiplying:
ⓐ 0÷20÷2 ⓑ 17/017/0
Divide. Then check by multiplying:
ⓐ 0÷60÷6 ⓑ 13/013/0
When the divisor or the dividend has more than one digit, it is usually easier to use the 412412 notation. This process is called long division. Let’s work through the process by dividing 7878 by 3.3.
Divide the first digit of the dividend, 7, by the divisor, 3. 

The divisor 3 can go into 7 two times since 2×3=62×3=6. Write the 2 above the 7 in the quotient. 

Multiply the 2 in the quotient by 2 and write the product, 6, under the 7. 

Subtract that product from the first digit in the dividend. Subtract 7−67−6. Write the difference, 1, under the first digit in the dividend. 

Bring down the next digit of the dividend. Bring down the 8. 

Divide 18 by the divisor, 3. The divisor 3 goes into 18 six times.


Write 6 in the quotient above the 8. 

Multiply the 6 in the quotient by the divisor and write the product, 18, under the dividend. Subtract 18 from 18. 

We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.
Check by multiplying the quotient times the divisor to get the dividend. Multiply 26×326×3 to make sure that product equals the dividend, 78.78.
216×3___78✓216×3___78✓
(4)It does, so our answer is correct.
 Step 1. Divide the first digit of the dividend by the divisor.
If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.  Step 2. Write the quotient above the dividend.
 Step 3. Multiply the quotient by the divisor and write the product under the dividend.
 Step 4. Subtract that product from the dividend.
 Step 5. Bring down the next digit of the dividend.
 Step 6. Repeat from Step 1 until there are no more digits in the dividend to bring down.
 Step 7. Check by multiplying the quotient times the divisor.
Divide 2,596÷4.2,596÷4. Check by multiplying:
Let's rewrite the problem to set it up for long division. 

Divide the first digit of the dividend, 2, by the divisor, 4. 

Since 4 does not go into 2, we use the first two digits of the dividend and divide 25 by 4. The divisor 4 goes into 25 six times. 

We write the 6 in the quotient above the 5. 

Multiply the 6 in the quotient by the divisor 4 and write the product, 24, under the first two digits in the dividend. 

Subtract that product from the first two digits in the dividend. Subtract 25−2425−24. Write the difference, 1, under the second digit in the dividend. 

Now bring down the 9 and repeat these steps. There are 4 fours in 19. Write the 4 over the 9. Multiply the 4 by 4 and subtract this product from 19. 

Bring down the 6 and repeat these steps. There are 9 fours in 36. Write the 9 over the 6. Multiply the 9 by 4 and subtract this product from 36. 

So 2,596÷4=6492,596÷4=649. 

Check by multiplying.


It equals the dividend, so our answer is correct.
Divide. Then check by multiplying: 2,636÷42,636÷4
Divide. Then check by multiplying: 2,716÷42,716÷4
Divide 4,506÷6.4,506÷6. Check by multiplying:
Divide. Then check by multiplying: 4,305÷5.4,305÷5.
Divide. Then check by multiplying: 3,906÷6.3,906÷6.
Divide 7,263÷9.7,263÷9. Check by multiplying.
Divide. Then check by multiplying: 4,928÷7.4,928÷7.
Divide. Then check by multiplying: 5,663÷7.5,663÷7.
So far all the division problems have worked out evenly. For example, if we had 2424 cookies and wanted to make bags of 88 cookies, we would have 33 bags. But what if there were 2828 cookies and we wanted to make bags of 8?8? Start with the 2828 cookies as shown in Figure 2.
Try to put the cookies in groups of eight as in Figure 3.
There are 33 groups of eight cookies, and 44 cookies left over. We call the 44 cookies that are left over the remainder and show it by writing R4 next to the 3.3. (The R stands for remainder.)
To check this division we multiply 33 times 88 to get 24,24, and then add the remainder of 4.4.
3×8___24+4___283×8___24+4___28
Divide 1,439÷4.1,439÷4. Check by multiplying.
Divide. Then check by multiplying: 3,812÷8.3,812÷8.
476 with a remainder of 4
Divide. Then check by multiplying: 4,319÷8.4,319÷8.
539 with a remainder of 7
Divide and then check by multiplying: 1,461÷13.1,461÷13.
Divide. Then check by multiplying: 1,493÷13.1,493÷13.
Divide. Then check by multiplying: 1,461÷12.1,461÷12.
Divide and check by multiplying: 74,521÷241.74,521÷241.
Let's rewrite the problem to set it up for long division. 
24174,52124174,521 
First we try to divide 241 into 7. Since that won’t work, we try 241 into 74. That still won’t work, so we try 241 into 745. Since 2 divides into 7 three times, we try 3. Since 3×241=7233×241=723, we write the 3 over the 5 in 745. Note that 4 would be too large because 4×241=9644×241=964, which is greater than 745. 

Multiply the 3 by 241 and subtract this product from 745. 

Now bring down the 2 and repeat these steps. 241 does not divide into 222. We write a 0 over the 2 as a placeholder and then continue. 

Now bring down the 1 and repeat these steps. Try 9. Since 9×241=2,1699×241=2,169, we write the 9 over the 1. Multiply the 9 by 241 and subtract this product from 2,221. 

There are no more numbers to bring down, so we are finished. The remainder is 52. So 74,521÷24174,521÷241 is 309 with a remainder of 52. 

Check by multiplying.
 
Sometimes it might not be obvious how many times the divisor goes into digits of the dividend. We will have to guess and check numbers to find the greatest number that goes into the digits without exceeding them.
Divide. Then check by multiplying: 78,641÷256.78,641÷256.
Divide. Then check by multiplying: 76,461÷248.76,461÷248.