Training Whole Numbers Multiplication and Division
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Multiplication and Division

20 min Whole Numbers

Multiplication and Division of Whole Numbers

Multiplication and division extend your arithmetic toolkit to handle repeated addition and equal sharing. Multiplication lets you scale quantities quickly, while division lets you split them into equal parts or determine how many times one number fits inside another.

This lesson covers the standard long-multiplication and long-division algorithms step by step, including how to handle zeros in the middle of a number. You will also learn about remainders and what they mean in real-world contexts.

Understanding multiplication and division thoroughly is essential because these operations appear everywhere — in fractions, proportions, area formulas, and algebraic equations.

Multiplication

Multiplication is repeated addition: $4 \times 3$ means $3 + 3 + 3 + 3 = 12$.

Properties of Multiplication
  • Commutative: $a \times b = b \times a$
  • Associative: $(a \times b) \times c = a \times (b \times c)$
  • Distributive: $a \times (b + c) = a \times b + a \times c$
  • Identity: $a \times 1 = a$
  • Zero Property: $a \times 0 = 0$

Multi-Digit Multiplication

  1. Multiply by the ones digit → write the partial product
  2. Multiply by the tens digit → shift one place left
  3. Continue for each digit, then add all partial products
Example 1

Multiply: $347 \times 26$

  1. $347 \times 6 = 2{,}082$
  2. $347 \times 20 = 6{,}940$
  3. Add: $2{,}082 + 6{,}940 = 9{,}022$

$$347 \times 26 = 9{,}022$$

Example 2

Multiply: $508 \times 403$

  1. $508 \times 3 = 1{,}524$
  2. $508 \times 400 = 203{,}200$
  3. $1{,}524 + 203{,}200 = 204{,}724$

$$508 \times 403 = 204{,}724$$

Division

Division splits a quantity into equal groups.

Definition

$$\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}$$

The remainder must satisfy $0 \le \text{Remainder} < \text{Divisor}$.

Example 3

Divide: $1{,}854 \div 6$

  1. $6$ into $18 = 3$. Write $3$.
  2. $6$ into $5 = 0$ R $5$. Write $0$, bring down $4$.
  3. $6$ into $54 = 9$.

$$1{,}854 \div 6 = 309$$

Check: $309 \times 6 = 1{,}854$ ✓

Example 4

Divide: $7{,}423 \div 15$

  1. $15$ into $74 = 4$ R $14$.
  2. Bring down $2$: $15$ into $142 = 9$ R $7$.
  3. Bring down $3$: $15$ into $73 = 4$ R $13$.

$$7{,}423 \div 15 = 494 \text{ R } 13$$

Check: $494 \times 15 + 13 = 7{,}423$ ✓

Example 5

Use the distributive property: $8 \times 97$

$$8 \times 97 = 8 \times (100 - 3) = 800 - 24 = 776$$

Important

Division by zero is undefined. There is no number that, when multiplied by $0$, gives a nonzero result.

Practice Problems

1. $234 \times 56$
2. $1{,}296 \div 8$
3. $705 \times 38$
4. $4{,}032 \div 12$
5. $999 \times 7$
6. $8{,}645 \div 25$
7. $1{,}024 \times 32$
8. $9{,}000 \div 125$
9. $460 \times 250$
10. $5{,}678 \div 14$
11. A box contains 24 packets. Each packet has 36 items. How many items total?
12. $12{,}000$ baseball cards shared equally among $16$ collectors. How many each?
13. $307 \times 204$
14. $15{,}625 \div 125$
15. Find the quotient and remainder: $2{,}000 \div 13$
Show Answer Key

1. $13{,}104$

2. $162$

3. $26{,}790$

4. $336$

5. $6{,}993$

6. $345$ R $20$

7. $32{,}768$

8. $72$

9. $115{,}000$

10. $405$ R $8$

11. $24 \times 36 = 864$ items

12. $750$ cards each

13. $62{,}628$

14. $125$

15. $153$ R $11$

Addition and Subtraction Order of Operations (PEMDAS)