Divisibility Rule of 105
2026-02-28 17:46 Diff
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Last updated on August 5, 2025

The divisibility rule is a way to find out whether a number is divisible by another number without using the division method. In real life, we can use the divisibility rule for quick math, dividing things evenly, and sorting things. In this topic, we will learn about the divisibility rule of 105.

What is the Divisibility Rule of 105?

The divisibility rule for 105 is a method by which we can find out if a number is divisible by 105 or not without using the division method. A number is divisible by 105 if it is divisible by 3, 5, and 7 (since 105 = 3 × 5 × 7). 

Check whether 315 is divisible by 105 using the divisibility rule.

Step 1: Check divisibility by 3. The sum of the digits is 3 + 1 + 5 = 9, which is divisible by 3.


Step 2: Check divisibility by 5. The last digit is 5, which means it is divisible by 5.


Step 3: Check divisibility by 7. Double the last digit and subtract it from the rest of the number: 31 - (5 × 2) = 21, which is a multiple of 7.

Since 315 is divisible by 3, 5, and 7, it is divisible by 105.
 

Tips and Tricks for Divisibility Rule of 105

Learning the divisibility rules will help kids master division. Let’s learn a few tips and tricks for the divisibility rule of 105.

  • Know the multiples of 105: Memorize the multiples of 105 (105, 210, 315, 420, etc.) to quickly check divisibility. If a number meets all divisibility rules for 3, 5, and 7, it is divisible by 105.
     
  • Use the divisibility rules for 3, 5, and 7: Ensure you are familiar with each rule separately to apply them effectively for 105.
     
  • Repeat the process for large numbers: Students should keep repeating the divisibility process for each component (3, 5, and 7) until they confirm divisibility by all three.
     
  • Use the division method to verify: Students can use the division method as a way to verify and cross-check their results. This will help them verify and also learn.
     

Common Mistakes and How to Avoid Them in Divisibility Rule of 105

The divisibility rule of 105 helps us quickly check if a given number is divisible by 105, but common mistakes like calculation errors can lead to incorrect results. Here, we will understand some common mistakes and how to avoid them.

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Problem 1

A shipment contains 525 boxes. Is the total number of boxes divisible by 105?

Okay, lets begin

Yes, 525 is divisible by 105.

Explanation

To check the divisibility of 525 by 105, we need to confirm divisibility by 3, 5, and 7.

 
1) Check divisibility by 3: Sum of digits is 5 + 2 + 5 = 12, which is divisible by 3.


2) Check divisibility by 5: The last digit is 5, so it is divisible by 5.


3) Check divisibility by 7: Multiply the last digit by 2, 5 × 2 = 10. Subtract from the rest, 52 – 10 = 42, which is divisible by 7.


Since 525 is divisible by 3, 5, and 7, it is divisible by 105.

Well explained 👍

Problem 2

A farmer has 630 apples and wants to pack them equally into boxes that can hold a number of apples divisible by 105. Can he do this?

Okay, lets begin

Yes, 630 apples can be packed equally into boxes divisible by 105.

Explanation

For 630 to be divisible by 105, it must be divisible by 3, 5, and 7.


1) Divisibility by 3: Sum of digits is 6 + 3 + 0 = 9, which is divisible by 3.


2) Divisibility by 5: The last digit is 0, so it is divisible by 5.


3) Divisibility by 7: Multiply the last digit by 2, 0 × 2 = 0. Subtract from the rest, 63 – 0 = 63, which is divisible by 7.
Since 630 is divisible by these three numbers, it is divisible by 105.

Well explained 👍

Problem 3

A concert hall has 840 seats. Are the seats arranged in rows that are divisible by 105?

Okay, lets begin

Yes, 840 seats can be arranged in rows divisible by 105.

Explanation

To check if 840 is divisible by 105, confirm divisibility by 3, 5, and 7.


1) Divisibility by 3: Sum of digits is 8 + 4 + 0 = 12, which is divisible by 3.


2) Divisibility by 5: The last digit is 0, so it is divisible by 5.


3) Divisibility by 7: Multiply the last digit by 2, 0 × 2 = 0. Subtract from the rest, 84 – 0 = 84, which is divisible by 7.
Thus, 840 is divisible by 105.
 

Well explained 👍

Problem 4

A company manufactures 945 widgets each week. Is this weekly production divisible by 105?

Okay, lets begin

Yes, 945 is divisible by 105.

Explanation

 For 945 to be divisible by 105, it must be divisible by 3, 5, and 7.


1) Divisibility by 3: Sum of digits is 9 + 4 + 5 = 18, which is divisible by 3.


2) Divisibility by 5: The last digit is 5, so it is divisible by 5.


3) Divisibility by 7: Multiply the last digit by 2, 5 × 2 = 10. Subtract from the rest, 94 – 10 = 84, which is divisible by 7.


Therefore, 945 is divisible by 105.

Well explained 👍

Problem 5

There are 1,155 chairs to be set up in a conference hall. Can the chairs be arranged in sections divisible by 105?

Okay, lets begin

Yes, 1,155 can be arranged in sections divisible by 105.

Explanation

To determine if 1,155 is divisible by 105, we check divisibility by 3, 5, and 7.


1) Divisibility by 3: Sum of digits is 1 + 1 + 5 + 5 = 12, which is divisible by 3.


2) Divisibility by 5: The last digit is 5, so it is divisible by 5.


3) Divisibility by 7: Multiply the last digit by 2, 5 × 2 = 10. Subtract from the rest, 115 – 10 = 105, which is divisible by 7.


Thus, 1,155 is divisible by 105.

Well explained 👍

FAQs on Divisibility Rule of 105

1.What is the divisibility rule for 105?

The divisibility rule for 105 requires checking if a number is divisible by 3, 5, and 7.

2.How many numbers are there between 1 and 1000 that are divisible by 105?

There are 9 numbers that can be divided by 105 between 1 and 1000. The numbers are 105, 210, 315, 420, 525, 630, 735, 840, and 945.

3.Is 420 divisible by 105?

 Yes, because 420 is divisible by 3, 5, and 7.

4.What if I meet only two of the divisibility rules?

If you meet only two of the divisibility rules, the number is not divisible by 105.
 

5.Does the divisibility rule of 105 apply to all integers?

Yes, the divisibility rule of 105 applies to all integers.

Important Glossaries for Divisibility Rule of 105

  • Divisibility rule: A set of rules used to determine if a number is divisible by another number without performing division.
     
  • Multiple: A product obtained by multiplying a number by an integer. For example, multiples of 105 are 105, 210, 315, etc.
     
  • Integer: A whole number that can be positive, negative, or zero.
     
  • Subtraction: The process of finding the difference between numbers by removing one from another.
     
  • Verification: The process of confirming the accuracy of a result, often by different means such as division.
     

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.