3-Digit Multiplication

3-digit multiplication in mathematics is a process of multiplying 3-digit numbers by 2-digit numbers, 1-digit numbers, or 3-digit numbers by placing numbers in columns according to their place values. Three-digit multiplication goes a step ahead if it is compared to 2-digit or 1-digit multiplication.

In this article, we will learn 3-digit by 1-digit multiplication, 3-digit by 2-digit multiplication, and 3-digit by 3-digit multiplication and solve a few examples for a better understanding of the concept.

3-digit multiplication is a method of multiplying 3-digit numbers with other numbers. When we multiply three-digit numbers, we arrange the numbers in columns according to the place values of the digits. We know that 3-digit numbers are arranged as per their place values as ones, tens, and hundreds. Once we have a set of two numbers to multiply, we usually keep the larger number on top and the smaller number below it. The number that is placed on top becomes the multiplicand and the number written below is the multiplier. When numbers are arranged according to their place values, we multiply the multiplier with all the digits of the multiplicand one by one starting from the ones digit, followed by the tens digit, and then the hundreds digit. All these products are written together and they result in the final product.

For example, if we need to multiply 123 × 3, we place them as shown below where 123 is the multiplicand and 3 is the multiplier. After multiplying these numbers we get the product as 269

Let us now learn how to do 3-digit multiplication with different numbers.

When a 3-digit number is multiplied by a 1-digit number, we have two scenarios.

• The first one refers to the multiplication in which the 1-digit number is simply multiplied by the 3-digit number without any carry-overs and we get the product. This is 3-digit multiplication without regrouping.
• The second one refers to the multiplication in which we multiply the 3-digit number with a 1-digit number and we need to carry over the extra digit of the product to the next column. This is 3-digit multiplication with regrouping. Let us discuss both the cases with the help of examples.

3-Digit Multiplication Without Regrouping

In order to find the product of a 3-digit number and a 1-digit number, we multiply the 1-digit number by each digit of the 3-digit number. If the product of the 1-digit number with each digit of the number is a single digit, then there is no need for carrying over any number. Let us consider an example.

Example: Multiply 214 × 2

Solution: The following steps show the procedure of multiplying 214 by 2.

• Step 1: Arrange the numbers 214 and 2 in columns according to their place values as shown in the figure given below.
• Step 2: Now, first we multiply the 1-digit number (2) by each digit of the 3-digit number (214)
  • When 2 is multiplied by 4, we get 8.
  • When 2 is multiplied by 1, we get 2.
  • When 2 is multiplied by 2, we get 4.
• Step 3: Therefore, the product that we get is 428.

3 Digit Multiplication With Regrouping

In this section, we will multiply a 3-digit number by a 1-digit number and see how regrouping works. Let us solve an example to demonstrate this.

Example: Multiply 347 by 3.

Solution: Let us multiply 347 by 3 using the steps given below.

• Step 1: Arrange the numbers 347 and 3 in columns according to their place values as shown below.
• Step 2: Multiply 3 by each digit of 347.
  • When 3 is multiplied by 7, we get 21. Since 21 is a 2-digit number, we write 1 under the ones column and carry 2 to the tens column above 4.
  • When 3 is multiplied by 4, we get 12. Now, we need to add the carry-over (2) to 12 and we get 14. Since 14 is a 2-digit number, we write 4 under the tens column and carry 1 to the hundreds column above 3.
  • When 3 is multiplied by 3, we get 9. Now, we need to add the carry-over 1 to 9 and we get 10. Since there is no other digit left for multiplication, we write 10.
• Step 3: Therefore, we get the product as 1041.

In order to multiply 3-digit numbers by 2-digit numbers, we first write the 3-digit number on top and the 2-digit number below it. Let us discuss the 3-digit by 2-digit multiplication without regrouping and with regrouping in the following sections.

3-Digit by 2-Digit Multiplication Without Regrouping

When we multiply a 3-digit number by a 2-digit number, we multiply the ones digit of the multiplier with the multiplicand, then we multiply the tens digit of the multiplier with the multiplicand. Then we add both these products to get the final product. Let us discuss the process step-by-step with the help of the following example.

Example: Multiply 411 by 31.

Solution: Let us multiply 411 by 31 stepwise.

• Step 1: Arrange the numbers 411 and 31 in columns according to their place values as shown below.
• Step 2: Multiply 1 by each digit of 411.
  • When 1 is multiplied by 1, we get 1.
  • When 1 is multiplied by 1, we get 1.
  • When 1 is multiplied by 4, we get 4. So, we have 411 as the first partial product.
• Step 3: Now, we place a zero under the first partial product, that is, just before we write the second partial product in the next line. This 0 is placed here because in this step we are actually multiplying 411 by 30.
• Step 4: Multiply 3 by each digit of 411.
  • When 3 is multiplied by 1, we get 3.
  • When 3 is multiplied by 1, we get 3.
  • When 3 is multiplied by 4, we get 12. So, we have 12330 as the second partial product.
• Step 5: Add these products to obtain the final answer.
• Step 6: 411 + 12330 = 12741. Therefore, the final product is 12741.

3-Digit by 2-Digit Multiplication With Regrouping

Now that we have multiplied a 3-digit number by a 2-digit number, let us try solving another problem involving regrouping or carrying.

Example: Multiply 573 by 46.

Solution: Let us multiply 573 by 46 using the following steps:

• Step 1: Arrange the numbers 573 and 46 in columns according to their place values as shown below.
• Step 2: Multiply 6 by each digit of 573.
  • When 6 is multiplied by 3, we get 18. Since 18 is a 2-digit number, we write 8 under the ones column and carry 1 to the tens column above 7.
  • When 6 is multiplied by 7, we get 42. Now, we need to add the carry-over (1) to 42 and we get 43. Since 43 is a 2-digit number, we write 3 in the tens column and carry 4 to the hundreds column above 5.
  • When 6 is multiplied by 5, we get 30. Now, we will add the carry-over (4) to 30, we get 34. Since there is no other digit left for multiplication, we write 34. So, we have 3438 in the first line (partial product) of the answer.
• Step 3: Now, we will place a zero under the first partial product, that is, before writing the second partial product in the next line. This is because in this step we are actually multiplying 573 with 40.
• Step 4: Multiply 4 by each digit of 573.
  • When 4 is multiplied by 3, we get 12. Since 12 is a 2-digit number, we write 2 under the tens column and carry 1 to the tens column above 7.
  • When 4 is multiplied by 7, we get 28. Now, we will add the carry-over 1 to 28 to get 29. Since 29 is a 2-digit number, we write 9 under the hundreds column and carry 2 to the hundreds column above 5.
  • When 4 is multiplied by 5, we get 20. Now, we will add the carried-over number 2 to 20 and we get 22. Since there is no other digit left for multiplication, we write 22. So, we have 22920 as the second line of the product.
• Step 5: Add these partial products to obtain the final answer.
• Step 6: This means 3438 + 22920 = 26358. Therefore, the final product is 26358.

In this section, we will learn how to multiply a 3-digit number by a 3-digit number. This process is similar to what we discussed in the previous sections. Let us understand 3-digit by 3-digit multiplication with the help of the following example.

Example: Multiply 123 by 456.

Solution: Let us multiply 123 by 456 using the following steps.

• Step 1: Arrange the numbers 123 and 456 in columns according to their place values as shown below.
• Step 2: Multiply 6 by each digit of 123.
  • When 6 is multiplied by 3, we get 18. Since 18 is a 2-digit number, we write 8 under ones column and carry 1 to the tens column above 2.
  • When 6 is multiplied by 2, we get 12. Now, we add the carried-over 1 to 12 and we get 13. Since 13 is a 2-digit number, we write 3 under the tens column and carry 1 to the next column above 1.
  • When 6 is multiplied by 1, we get 6. Now, add the carried-over 1 to 6 to get 7. Since there is no other digit left for multiplication, we write 7. So, we have 738 in the first line as the partial product.
• Step 3: Now, place a zero under this partial product under ones column. This is because in this step we are actually multiplying 123 with 50.
• Step 4: Multiply 5 by each digit of 123.
  • When 5 is multiplied by 3, we get 15. Since 15 is a 2-digit number, we write 5 in the tens column and carry 1 to the next column above 2.
  • When 5 is multiplied by 2, we get 10. Now, add the carried-over 1 to 10 to get 11. Since 11 is a 2-digit number, we write 1 in hundreds column and carry 1 to the next column above 1.
  • When 5 is multiplied by 1, we get 5. Now, add the carried-over 1 to 5 to get 6. Since there is no other digit left for multiplication, we write 6. So, we have 6150 in the second line of the partial product.
• Step 5: Now, place two zeros (0s) under the ones and tens column under the partial product obtained in the previous step. This is because in this step we are actually multiplying 123 with 400.
• Step 6: Multiply 4 by each digit of 123.
  • When 4 is multiplied by 3, we get 12. Since 12 is a 2-digit number, we write 2 under the hundreds column and carry 1 to the next column above 2.
  • When 4 is multiplied by 2, we get 8. Now, add the carried-over 1 to 8 to get 9. We write 9 under the next column.
  • When 4 is multiplied by 1, we get 4. Since there is no other digit left for multiplication, we write 4. So, we have 49200 in the third line as the partial product.
• Step 7: Add all the 3 partial products to obtain the final product. This means 738 + 6150 + 49200 = 56088.
• Step 8: Therefore, the final product is 56088.

☛ Related Topics

• 2-Digit Subtraction
• 2-Digit Addition
• 3-Digit Addition
• 3-Digit Subtraction
• 2-Digit Multiplication
• 4-Digit Addition
• 4-Digit Subtraction
• Multiplication and Division of Integers

What is 3-Digit Multiplication?

3-digit multiplication in mathematics is a process of multiplying 3-digit numbers by 1-digit numbers, 2-digit numbers, and 3-digit numbers by placing the numbers in columns according to their place values.

How to do 3-Digit Multiplication?

3-digit multiplication can be done easily if the numbers are arranged according to their place values. Once we have a set of two numbers to multiply, we usually keep the larger number on top and the smaller number below it. The number that is placed on top becomes the multiplicand and the number written below is the multiplier. When numbers are arranged according to their place values, we multiply the multiplier with all the digits of the multiplicand one by one starting from the ones digit, followed by the tens digit, and then the hundreds digit. All these products are written together and they result in the final product.

How to do 3-Digit by 1-Digit Multiplication?

To multiply a 3-digit number by a 1-digit number, we multiply the 1-digit number by each digit of the 3-digit number to get the product. For example, let us multiply 314 × 2. We multiply 2 with 4 to get 8 which will be placed under the ones column. Then, we will multiply 2 with 1 to get 2 which will be placed under the tens column. After this, we will multiply 2 with 3 to get 6. Therefore, the product of 314 × 2 = 628.

What is 3-Digit by 2-Digit Multiplication?

When a 3-digit number is multiplied by a 2-digit number, we multiply each digit of the 2-digit number by each digit of the 3-digit number. We arrange the numbers in columns according to their place values, write the partial products one below the other and add them to get the final product.

What is 3-Digit by 3-Digit Multiplication?

3-digit by 3-digit multiplication means when a 3-digit number is multiplied with another 3-digit number. It is done by placing the numbers column-wise and then multiplying each digit of one number by each digit of the other number. The partial products are written one below the other and then the products are added to get the final answer.