Division of Whole Numbers

What You’ll Learn

In this lesson you’ll start with what division really means using small numbers, then build up to the long division algorithm for bigger problems.

The Basics

Division means splitting something into equal groups, or figuring out how many times one number fits inside another. It’s the opposite of multiplication.

If you have 12 cookies and want to share them equally among 3 friends, each person gets $12 \div 3 = 4$ cookies. You can check: $4 \times 3 = 12$. That’s division, finding the size of each group.

Here are a few single-digit examples:

• $8 \div 2 = 4$ - eight split into two groups of four
• $15 \div 5 = 3$ - five goes into fifteen three times
• $18 \div 3 = 6$
• $24 \div 6 = 4$

Some handy patterns:

• Anything divided by 1 stays the same: $9 \div 1 = 9$
• Anything divided by itself is 1: $7 \div 7 = 1$
• Zero divided by anything is 0: $0 \div 5 = 0$
• You can never divide by 0. It’s undefined (there’s no way to split something into zero groups)

Division and multiplication are inverses. If you know $6 \times 8 = 48$, then you also know $48 \div 8 = 6$ and $48 \div 6 = 8$. Your multiplication facts double as division facts.

Remainders happen when a number doesn’t split evenly. If you have 13 cookies for 4 people, each gets 3 with 1 left over: $13 \div 4 = 3$ remainder $1$. That leftover is the remainder.

Long Division

Once you’re comfortable with basic division facts, the same idea scales up to bigger numbers using the long division algorithm. It’s a step-by-step process: Divide, Multiply, Subtract, Bring down.

The steps:

1. Divide: How many times does the divisor go into the first digits?
2. Multiply: Multiply the quotient digit by the divisor.
3. Subtract: Subtract from the current partial dividend.
4. Bring down the next digit and repeat.
5. Remainder: Whatever is left over if the division isn’t exact.

Example: $456 \div 6$

$$\begin{array}{r} 76 \\ 6\overline{)456} \\ \underline{-42}\phantom{6} \\ 36 \\ \underline{-36} \\ 0 \end{array}$$

• 6 into 45 = 7 → $7 \times 6 = 42$, subtract → 3
• Bring down 6 → 36
• 6 into 36 = 6 → $6 \times 6 = 36$, subtract → 0

Quotient: $76$, remainder $0$.

If there’s a remainder: $457 \div 6 = 76$ remainder $1$ (because $6 \times 76 + 1 = 457$).

Check: Quotient $\times$ divisor $+$ remainder $=$ dividend.

Worked Examples

Divide $8{,}736 \div 24$

$$\begin{array}{r} 364 \\ 24\overline{)8736} \\ \underline{-72}\phantom{36} \\ 153\phantom{6} \\ \underline{-144}\phantom{6} \\ 96 \\ \underline{-96} \\ 0 \end{array}$$

1. 24 into 87 = 3 → $3 \times 24 = 72$, subtract → 15
2. Bring down 3 → 153
3. 24 into 153 = 6 → $6 \times 24 = 144$, subtract → 9
4. Bring down 6 → 96
5. 24 into 96 = 4 → $4 \times 24 = 96$, subtract → 0

Quotient: $364$, remainder $0$.

Check: $364 \times 24 = 8{,}736$ ✓

Real-World Applications

Division figures shares, rates, or quantities: Split 840 dollars among 4 people = 210 dollars each; miles per gallon ($420$ miles $\div$ $15$ gallons $= 28$ mpg); or items per box ($96$ cookies $\div$ $12 = 8$ per box). It answers “how many” or “how much each” questions in budgeting, work, or planning.

You’ve Got This

Long division is a step-by-step process. Take it one digit at a time, guess conservatively, and always check by multiplying back. If remainders confuse you, just remember they mean “left over.” Practice on paper and you’ll master it quickly.

Quiz

What is 864 ÷ 12?

• A.82
• B.70
• C.72
• D.74

Divide 500 ÷ 8. What is the quotient and remainder?

• A.60 r 4
• B.63 r 0
• C.62 r 4
• D.62 r 6

A total bill of 240 dollars is split equally among 6 friends. How much does each person pay?

• A.45 dollars
• B.40 dollars
• C.50 dollars
• D.30 dollars

What is 1,575 ÷ 25?

• A.65
• B.60
• C.70
• D.63

What is 945 ÷ 9?

• A.105
• B.95
• C.115
• D.100