In-Depth: Divisibility Rules
CLAT Application & Relevance
Importance: Low. While not directly tested as standalone questions, a quick knowledge of divisibility rules can significantly speed up calculations in Data Interpretation, especially when dealing with large numbers or when simplifying fractions and ratios. It's a foundational skill that boosts calculation efficiency.
How it's tested: Indirectly, by allowing faster reduction of fractions in ratios, quicker checking of totals in tables, or in number-based logical reasoning scenarios (e.g., finding a missing digit in a number that is known to be divisible by 3 and 9).
Section 1: Core Concepts & Rules
Divisibility rules are shortcuts to determine if a number is evenly divisible by another number without performing long division. This is useful for simplification and factorisation.
Key Divisibility Rules (from 2 to 11)
- Divisibility by 2: A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, 8).
- Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
- Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
- Divisibility by 7: (More complex, often not worth the time for large numbers; direct division is faster). Double the last digit and subtract it from the remaining number. If the result is divisible by 7, the original number is too. (e.g., 343: 34 - (2*3) = 34 - 6 = 28. 28 is divisible by 7, so 343 is.)
- Divisibility by 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
- Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. (Note: If divisible by 9, it's also divisible by 3).
- Divisibility by 10: A number is divisible by 10 if its last digit is 0.
- Divisibility by 11: A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or a multiple of 11. (e.g., 121: (1+1)-2 = 0. So 121 is divisible by 11. For 1331: (1+3)-(3+1)=0. So 1331 is divisible by 11).
Section 2: Solved CLAT-Style Examples (Application)
Example 1: Using Divisibility Rules for Data Validation (Caselet)
Passage Context: "A law firm is assigning unique client IDs. A new rule states that for certain sensitive cases, the 5-digit client ID must be divisible by both 3 and 5. One such ID is 4X750, where 'X' is a missing digit."
Question: "What is the smallest possible value of 'X' that makes the client ID divisible by both 3 and 5?"
Detailed Solution:
1. Divisibility by 5: The last digit is 0, so 4X750 is already divisible by 5, regardless of 'X'.
2. Divisibility by 3: The sum of the digits must be divisible by 3.
Sum of digits = 4 + X + 7 + 5 + 0 = 16 + X.
3. Find smallest 'X' for (16 + X) to be divisible by 3:
- If X = 0, Sum = 16 (not divisible by 3).
- If X = 1, Sum = 17 (not divisible by 3).
- If X = 2, Sum = 18 (divisible by 3).
Answer: The smallest possible value of 'X' is 2. (The client ID would be 42750).
Example 2: Applying Divisibility for Problem Solving (DI Context)
Passage Context: "A company's quarterly production targets are given. For Quarter 1, the target was 12,345 units. For Quarter 2, it was 23,456 units. For Quarter 3, the target was 34,578 units. The quality control department states that only production batches perfectly divisible by 9 are considered 'high-standard'."
Question: "Which quarter's production target (if any) was considered 'high-standard'?"
Detailed Solution:
1. Apply Divisibility Rule for 9 (Sum of digits must be divisible by 9):
Quarter 1 (12,345): Sum = 1+2+3+4+5 = 15. (15 is not divisible by 9).
Quarter 2 (23,456): Sum = 2+3+4+5+6 = 20. (20 is not divisible by 9).
Quarter 3 (34,578): Sum = 3+4+5+7+8 = 27. (27 is divisible by 9).
Answer: Quarter 3's production target (34,578 units) was considered 'high-standard'.