Definition: Divisibility rules are a set of mathematical shortcuts used to determine if a given integer is divisible by another without performing the complete process of long division. These rules are essential tools for competitive exam aspirants, as they significantly reduce calculation time in sections like Quantitative Aptitude and Logical Reasoning.
The Fundamentals of Divisibility
In competitive exams like the UPSC CSAT, time is your most valuable asset. While long division is accurate, it is often unnecessary. Divisibility rules allow you to break down large numbers into manageable parts, enabling you to identify factors, solve remainder problems, and simplify fractions almost instantly.
Most rules rely on the last digits of a number or the sum of its digits. By understanding the underlying base-10 structure of our number system, you can quickly verify if a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10, or 11. Mastering these does not just help in arithmetic; it is a prerequisite for solving complex Number System problems.
Divisibility by 2, 4, 5, 8, and 10 (The Power of Last Digits)
These rules focus exclusively on the tail end of a number. Because our system is based on powers of 10, these rules are derived from how many of the final digits we need to check.
- Divisibility by 2: The last digit must be even (0, 2, 4, 6, or 8).
- Divisibility by 4: The last two digits must form a number divisible by 4.
- Divisibility by 8: The last three digits must form a number divisible by 8.
- Divisibility by 5: The last digit must be either 0 or 5.
- Divisibility by 10: The last digit must be 0.
Teacher’s Tip: Think of 4 as 22 and 8 as 23. This is why you check the last 2 digits for 4, and the last 3 digits for 8. The higher the power, the more digits you examine.
Divisibility by 3, 6, and 9 (The Summation Method)
Unlike the previous rules, divisibility by 3 and 9 requires you to look at the sum of the digits of the number. If the sum of the digits is divisible by 3, the entire number is divisible by 3. Similarly, if the sum is divisible by 9, the entire number is divisible by 9.
The rule for 6 is a composite rule. Since 6 is the product of 2 and 3 (which are coprime), a number is divisible by 6 if and only if it passes the test for both 2 and 3. This means the number must be even AND the sum of its digits must be a multiple of 3.
Advanced Divisibility: The Rule of 11
The rule for 11 is distinct and often appears in UPSC CSAT questions involving large integers. To check for divisibility by 11, calculate the alternating sum of the digits. You subtract and add the digits starting from the right (or left) and check if the result is 0 or a multiple of 11 (including negative multiples like -11, -22, etc.).
For example, in the number 1,331: 1 – 3 + 3 – 1 = 0. Since the result is 0, 1,331 is perfectly divisible by 11. This method is highly effective for identifying patterns in sequences or checking the validity of large numerical codes.
Important Facts and Formulas
| Divisor | Rule |
|---|---|
| 2 | Last digit is even |
| 3 | Sum of digits is divisible by 3 |
| 4 | Last two digits divisible by 4 |
| 8 | Last three digits divisible by 8 |
| 9 | Sum of digits is divisible by 9 |
| 11 | Difference of alternating sum is 0 or multiple of 11 |
Previous Year Question Hints
Example 1: If a 3-digit number 4×2 is divisible by 3, what are the possible values of x?
Hint: Sum of digits (4+x+2) = 6+x. For this to be divisible by 3, x can be 0, 3, 6, or 9.
Example 2: A number is formed by writing a 2-digit number twice (e.g., 2525). Is it divisible by 11?
Hint: Use the alternating sum method: 5-2+5-2 = 6. Since 6 is not divisible by 11, the number 2525 is not divisible by 11.
Quick Revision Summary
- Even numbers are always divisible by 2.
- Divisibility by 3 and 9 depends entirely on the sum of the digits.
- Divisibility by 4 and 8 depends on the last 2 and 3 digits respectively.
- Divisibility by 6 requires the number to be both even and divisible by 3.
- Divisibility by 11 uses the alternating sum of digits.
- If a number is divisible by co-prime factors (e.g., 3 and 4), it is divisible by their product (12).
- Always look for the sum of digits first if the number is large; it is the fastest mental check.
- Practice these rules with PYQs to build speed and accuracy for the CSAT exam.
