Introduction to Divisibility Rules

In arithmetic, understanding whether one number can be divided evenly by another without leaving a remainder is a fundamental concept called divisibility. Divisibility rules are simple tests or shortcuts that help us quickly determine if a number is divisible by another number, without performing long division.

Why is divisibility important? It helps us:

• Find factors of numbers quickly
• Simplify fractions by identifying common factors
• Solve problems efficiently in competitive exams where time is limited

Before diving into the rules, let's briefly recall the number system hierarchy:

• Natural numbers: Counting numbers starting from 1, 2, 3, ...
• Whole numbers: Natural numbers including 0
• Integers: Whole numbers and their negatives, e.g., -3, -2, -1, 0, 1, 2, 3
• Rational numbers: Numbers expressed as fractions of integers
• Real numbers: All rational and irrational numbers

Divisibility rules mainly apply to integers and natural numbers. Mastering these rules will sharpen your number sense and speed up calculations.

Basic Divisibility Rules

We start with the simplest and most commonly used divisibility rules for 2, 3, and 5. These rules rely on patterns in the digits of numbers.

Let's understand why these rules work:

• Divisible by 2: Even numbers end with digits 0, 2, 4, 6, or 8. Since 10 is divisible by 2, the divisibility depends only on the last digit.
• Divisible by 3: Because 10 ≡ 1 (mod 3), each digit's place value contributes the same remainder as the digit itself modulo 3. So, summing digits and checking divisibility by 3 works.
• Divisible by 5: Since 10 ≡ 0 (mod 5), only the last digit determines divisibility by 5.

Intermediate Divisibility Rules

Now, let's move to slightly more complex rules for divisibility by 4, 6, and 9. These rules build on the basic ones and sometimes combine them.

Why these rules work:

• Divisible by 4: Since 100 is divisible by 4, only the last two digits affect divisibility by 4.
• Divisible by 6: 6 = 2 x 3, so a number must be divisible by both 2 and 3.
• Divisible by 9: Similar to 3, because 10 ≡ 1 (mod 9), sum of digits determines divisibility.

Advanced Divisibility Rules

Divisibility rules for 7, 11, and 13 are more involved but very useful in competitive exams. These rules often use subtraction, addition, or alternating sums.

Divisibility by 7

One popular method:

• Double the last digit of the number.
• Subtract this from the rest of the number.
• Repeat the process if needed.
• If the result is divisible by 7 (including 0), the original number is divisible by 7.

graph TD    A[Start with number n] --> B[Double the last digit]    B --> C[Subtract from remaining leading part]    C --> D{Is result small enough?}    D -- No --> B    D -- Yes --> E{Is result divisible by 7?}    E -- Yes --> F[Number divisible by 7]    E -- No --> G[Number not divisible by 7]

Divisibility by 11

Calculate the alternating sum of digits:

• Add digits in odd positions and subtract digits in even positions (from right to left or left to right consistently).
• If the result is divisible by 11 (including 0), the number is divisible by 11.

Divisibility by 13

Multiply the last digit by 4 and add it to the remaining leading part. Repeat if necessary. If the result is divisible by 13, so is the original number.