Definition: Divisibility rules are a set of mathematical shortcuts used to determine if a given integer is divisible by another integer without performing the complete process of long division. These rules are essential for competitive exams like the UPSC CSAT, as they allow aspirants to simplify complex calculations, identify factors, and solve number system problems with speed and precision.
The Foundation: Divisibility by 2, 4, 5, 8, and 10
In competitive examinations, the most frequently used rules involve powers of 2 and 5. These are based on the final digits of a number, which makes them incredibly efficient for quick mental calculations. By observing only the last few digits, you can eliminate options in multiple-choice questions rapidly.
For a number to be divisible by 2, the last digit must be an even number (0, 2, 4, 6, or 8). When checking for divisibility by 4, look at the last two digits of the number; if the number formed by these two digits is divisible by 4, the entire number is divisible by 4. Similarly, for divisibility by 8, we examine the last three digits.
A useful tip for 8: If the hundreds digit is even, you only need to check if the last two digits are divisible by 8. If the hundreds digit is odd, check if the last two digits plus 4 are divisible by 8.
- Divisibility by 5: The last digit must be 0 or 5.
- Divisibility by 10: The last digit must be 0.
The Summation Rules: Divisibility by 3 and 9
When dealing with 3 and 9, we shift our focus from the position of the digits to the sum of the digits. This is a powerful tool when you encounter large numbers or algebraic expressions involving variables that need to satisfy certain divisibility criteria.
For a number to be divisible by 3, the sum of all its digits must be divisible by 3. The rule for divisibility by 9 is even stricter: the sum of the digits must be a multiple of 9. If the sum of the digits is not a multiple of 9, the number is not divisible by 9, regardless of how large it is.
These rules are particularly helpful in solving digital root problems or finding missing digits in a number where a variable is given. Always remember that if a number is divisible by 9, it is automatically divisible by 3, but the converse is not necessarily true.
Advanced Rules: Divisibility by 6, 11, and Composite Numbers
When a divisor is a composite number, we use the co-prime factor method. A number is divisible by a composite divisor if it is divisible by each of its co-prime factors. For example, to check for divisibility by 6, the number must satisfy both the rule for 2 (it must be even) and the rule for 3 (the sum of digits must be divisible by 3).
The rule for divisibility by 11 is a favorite for UPSC examiners. To check if a number is divisible by 11, find the difference between the sum of the digits at odd positions and the sum of the digits at even positions. If this difference is 0 or a multiple of 11 (including negative multiples like -11, -22), the original number is divisible by 11.
- Rule for 7: Double the last digit and subtract it from the remaining number. If the result is divisible by 7, the original number is divisible by 7.
- Rule for 12: The number must be divisible by both 3 and 4.
- Rule for 15: The number must be divisible by both 3 and 5.
Important Facts and Formulas
| Divisor | Rule |
|---|---|
| 2 | Last digit is even |
| 3 | Sum of digits is divisible by 3 |
| 4 | Last 2 digits are divisible by 4 |
| 8 | Last 3 digits are divisible by 8 |
| 9 | Sum of digits is divisible by 9 |
| 11 | (Sum of odd pos digits) – (Sum of even pos digits) = 0 or multiple of 11 |
Previous Year Question Hints
Example 1: If a 3-digit number ‘abc’ is such that ‘a+b+c’ is divisible by 9, what can you conclude? Hint: The number itself is divisible by 9. This is a common logic used in UPSC CSAT to identify properties of numbers without knowing the exact value.
Example 2: A number is formed by repeating a 2-digit number three times (e.g., 252525). Is it divisible by 7, 11, or 13? Hint: Any number of the form ‘xyxyxy’ is divisible by 10101, which is 3 × 7 × 13 × 37. Recognizing these patterns saves significant time.
Quick Revision Summary
- Last Digit: Use for 2, 5, and 10.
- Last Two Digits: Use for 4 and 25.
- Sum of Digits: Use for 3 and 9.
- Alternating Sum: Use for 11.
- Composite Divisors: Break into co-prime factors (e.g., 6 = 2 × 3).
- Co-prime definition: Two numbers are co-prime if their HCF is 1.
- Zero Property: 0 is divisible by every non-zero integer.
- Practice: Apply these rules to past year papers to build intuition for when to use which rule.
