Multiplying Whole Numbers

Multiplication is repeated addition. $4 \times 3$ means “add 4 three times” ($4 + 4 + 4 = 12$) or equivalently “add 3 four times” ($3 + 3 + 3 + 3 = 12$). But for large numbers, you need a systematic method — long multiplication — that breaks the problem into simple single-digit multiplications and additions.

Single-Digit Multiplication Facts

Fluency with single-digit multiplication (the “times tables”) makes everything else faster. These are the facts worth memorizing:

Multiplying by Powers of 10

To multiply by 10, 100, 1000, etc., append the corresponding number of zeros:

$45 \times 10 = 450$ $45 \times 100 = 4{,}500$ $45 \times 1{,}000 = 45{,}000$

Long Multiplication (Multi-Digit)

When multiplying by a number with two or more digits, multiply by each digit separately, then add the partial products.

Example 1: Multiply $37 \times 4$

Multiply each digit of 37 by 4, right to left:

• Ones: $7 \times 4 = 28$ → write 8, carry 2
• Tens: $3 \times 4 = 12$, plus carried 2 = 14

Answer: $148$

Example 2: Multiply $253 \times 6$

• Ones: $3 \times 6 = 18$ → write 8, carry 1
• Tens: $5 \times 6 = 30$, plus 1 = 31 → write 1, carry 3
• Hundreds: $2 \times 6 = 12$, plus 3 = 15

Answer: $1{,}518$

Example 3: Multiply $47 \times 23$

This requires two partial products:

First partial product (multiply 47 by 3):

• $7 \times 3 = 21$ → write 1, carry 2
• $4 \times 3 = 12$, plus 2 = 14

First partial product: $141$

Second partial product (multiply 47 by 20):

Multiply 47 by 2, then shift one place left (add a 0):

• $7 \times 2 = 14$ → write 4, carry 1
• $4 \times 2 = 8$, plus 1 = 9

Second partial product: $940$

Add the partial products:

$141 + 940 = 1{,}081$

Answer: $1{,}081$

Example 4: Multiply $385 \times 27$

Multiply 385 by 7:

• $5 \times 7 = 35$ → write 5, carry 3
• $8 \times 7 = 56$, plus 3 = 59 → write 9, carry 5
• $3 \times 7 = 21$, plus 5 = 26

First partial product: $2{,}695$

Multiply 385 by 20:

• $5 \times 2 = 10$ → write 0, carry 1
• $8 \times 2 = 16$, plus 1 = 17 → write 7, carry 1
• $3 \times 2 = 6$, plus 1 = 7

Second partial product: $7{,}700$

Add:

$2{,}695 + 7{,}700 = 10{,}395$

Answer: $10{,}395$

Estimation: Checking Reasonableness

Before (or after) multiplying, estimate to make sure your answer is in the right ballpark:

• $385 \times 27 \approx 400 \times 25 = 10{,}000$

Our answer of 10,395 is close to 10,000, so it is reasonable.

Multiplication Properties

These properties make multiplication flexible:

Commutative Property: Order does not matter.

$a \times b = b \times a$

$7 \times 3 = 3 \times 7 = 21$

Associative Property: Grouping does not matter.

$(a \times b) \times c = a \times (b \times c)$

$(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24$

Distributive Property: Multiplication distributes over addition.

$a \times (b + c) = a \times b + a \times c$

$6 \times 14 = 6 \times (10 + 4) = 60 + 24 = 84$

This is actually what long multiplication does — it distributes the multiplication across each place value.

Identity Property: Any number times 1 equals itself.

$a \times 1 = a$

Zero Property: Any number times 0 equals 0.

$a \times 0 = 0$

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Multiplication is repeated addition, but long multiplication is the efficient method for large numbers
• Multiply by each digit separately, shifting left for each place value, then add the partial products
• Estimate first to check if your answer is reasonable
• Memorize the single-digit multiplication facts — they are the building blocks
• The distributive property is the reason long multiplication works

Return to Arithmetic for more foundational math topics.