Introduction to Divisibility Rules

In mathematics, divisibility is a fundamental concept that helps us understand when one number can be divided by another without leaving any remainder. For example, 20 is divisible by 5 because 20 / 5 = 4, which is a whole number. However, 20 is not divisible by 3 because 20 / 3 = 6.66..., which is not an integer.

Divisibility rules are simple tests or shortcuts that allow us to quickly determine whether a number is divisible by another number without performing the full division. These rules are especially useful in competitive exams where speed and accuracy are crucial. Instead of long division, you can use these rules to save time and avoid errors.

Understanding divisibility rules also lays the foundation for other important topics like finding factors, multiples, simplifying fractions, and solving problems involving the Least Common Multiple (LCM) and Highest Common Factor (HCF).

Basic Divisibility Rules

Let's start with the simplest and most commonly used divisibility rules. These rules rely mostly on the digits of the number and can be applied quickly.

Notice how these rules often depend on the last digit(s) or the sum of digits. This is because of the way our number system is structured in base 10.

Advanced Divisibility Rules

Some divisibility rules are a bit more complex but still manageable with practice. These include divisibility by 6, 7, and 11.

Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 and 3. This means:

• The last digit is even (divisible by 2), and
• The sum of the digits is divisible by 3.

Example: Check if 234 is divisible by 6.

• Last digit is 4 (even) -> divisible by 2
• Sum of digits = 2 + 3 + 4 = 9, which is divisible by 3
• Therefore, 234 is divisible by 6.

Divisibility by 7

Divisibility by 7 is less straightforward. One common method is:

1. Double the last digit of the number.
2. Subtract this doubled value from the rest of the number (the number formed by all digits except the last one).
3. If the result is divisible by 7 (including 0), then the original number is divisible by 7.
4. If not, repeat the process with the new number.

Divisibility by 11

To check divisibility by 11:

1. Add the digits in the odd positions (from right to left).
2. Add the digits in the even positions.
3. Find the difference between these two sums.
4. If the difference is 0 or divisible by 11, then the number is divisible by 11.

graph TD    A[Start] --> B{Divisibility by 7?}    B --> C[Double last digit]    C --> D[Subtract from rest of number]    D --> E{Result divisible by 7?}    E -->|Yes| F[Number divisible by 7]    E -->|No| G[Repeat with result]    G --> C    H[Start] --> I{Divisibility by 11?}    I --> J[Sum digits at odd positions]    I --> K[Sum digits at even positions]    J --> L[Find difference]    K --> L    L --> M{Difference divisible by 11?}    M -->|Yes| N[Number divisible by 11]    M -->|No| O[Not divisible by 11]

Worked Examples