Mastering Divisibility Rules and Properties for Competitive Exams
Welcome to the foundational module on Divisibility Rules and Properties, a cornerstone for success in the Quantitative Aptitude section of competitive exams like the CAT. Understanding these rules allows for rapid calculations, simplification of complex problems, and efficient identification of factors and multiples. This module will equip you with the essential knowledge to tackle number theory questions with confidence.
What are Divisibility Rules?
Divisibility rules are shortcuts that help us determine if a number is divisible by another number without performing the actual division. They are based on the properties of numbers and are incredibly useful for mental math and simplifying calculations in exams.
Key Divisibility Rules
Divisibility by 2, 5, and 10 are determined by the last digit.
A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8). A number is divisible by 5 if its last digit is 0 or 5. A number is divisible by 10 if its last digit is 0.
These rules are straightforward because our number system is base-10. For divisibility by 2, any number ending in an even digit is a multiple of 2. For divisibility by 5, numbers ending in 0 or 5 are multiples of 5. For divisibility by 10, a number must be a multiple of both 2 and 5, hence it must end in 0.
0
Divisibility by 3 and 9 depends on the sum of the digits.
A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 9 if the sum of its digits is divisible by 9.
This rule stems from the fact that any power of 10 leaves a remainder of 1 when divided by 3 or 9 (e.g., 10 = 33 + 1, 100 = 333 + 1, 1000 = 3*333 + 1). When you sum the digits of a number, you are essentially summing the remainders of each place value when divided by 3 or 9. Therefore, if the sum of digits is divisible by 3 (or 9), the entire number will be.
Yes, because the sum of its digits (1+2+3+4+5 = 15) is divisible by 3.
Divisibility by 4 and 8 relates to the last few digits.
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
This is because 100 is divisible by 4 (100 = 4 * 25) and 1000 is divisible by 8 (1000 = 8 * 125). Any number can be expressed as a sum of multiples of 100 (or 1000) plus the number formed by its last two (or three) digits. Since the multiples of 100 (or 1000) are already divisible by 4 (or 8), the divisibility of the entire number depends solely on the divisibility of the number formed by its last two (or three) digits.
The last two digits form 92. Since 92 is divisible by 4 (92 = 4 * 23), the number 567892 is divisible by 4.
Divisibility by 6 requires divisibility by both 2 and 3.
A number is divisible by 6 if it is divisible by both 2 and 3.
Since 6 is the product of two prime numbers (2 and 3), a number must be divisible by both of these primes to be divisible by 6. This means the number must be even (divisible by 2) and the sum of its digits must be divisible by 3.
Yes. 456 is even (divisible by 2). The sum of its digits is 4+5+6 = 15, which is divisible by 3. Therefore, 456 is divisible by 6.
Divisibility by 11 involves alternating sums of digits.
A number is divisible by 11 if the alternating sum of its digits (starting from the rightmost digit, adding the first, subtracting the second, adding the third, and so on) is divisible by 11.
Consider a number like abcde. This can be written as 10000a + 1000b + 100c + 10d + e. When we look at powers of 10 modulo 11: 10 = -1 (mod 11), 100 = 1 (mod 11), 1000 = -1 (mod 11), 10000 = 1 (mod 11). So, abcde = a(1) + b(-1) + c(1) + d(-1) + e(1) (mod 11). This simplifies to e - d + c - b + a (mod 11). If this alternating sum is divisible by 11, the original number is too.
Visualizing the alternating sum for divisibility by 11. For a number like 94827, we sum digits at odd positions (from the right) and subtract the sum of digits at even positions. Positions: 5 4 3 2 1. Digits: 9 4 8 2 7. Alternating sum: (7 + 8 + 9) - (2 + 4) = 24 - 6 = 18. Since 18 is not divisible by 11, 94827 is not divisible by 11. If the number was 13574, the alternating sum would be (4+5+1) - (7+3) = 10 - 10 = 0. Since 0 is divisible by 11, 13574 is divisible by 11.
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Alternating sum: (9 + 8 + 7) - (2 + 1) = 24 - 3 = 21. Since 21 is not divisible by 11, 71829 is not divisible by 11.
Properties of Divisibility
Beyond specific rules, understanding general properties of divisibility is crucial for solving more complex problems.
| Property | Description | Example |
|---|---|---|
| If a number 'a' divides 'b', and 'b' divides 'c', then 'a' divides 'c'. | Transitive Property of Divisibility | If 2 divides 6, and 6 divides 18, then 2 divides 18. |
| If a number 'a' divides 'b' and 'a' divides 'c', then 'a' divides their sum (b+c) and difference (b-c). | Sum and Difference Property | If 3 divides 12 and 3 divides 15, then 3 divides (12+15=27) and 3 divides (15-12=3). |
| If a number 'a' divides 'b', then 'a' divides 'kb' for any integer 'k'. | Multiplication Property | If 5 divides 20, then 5 divides (4*20=80). |
| If 'a' divides 'b' and 'b' divides 'a', then |a| = |b|. | Symmetric Property (for non-zero numbers) | If 7 divides 14 and 14 divides 7, this is not possible unless we consider negative numbers, e.g., if 7 divides -7 and -7 divides 7, then |7| = |-7|. |
Remember that these properties are fundamental. They are often implicitly used in algebraic manipulations and number theory problems.
Applying Divisibility Rules in Problem Solving
Divisibility rules are not just for checking numbers; they are powerful tools for simplification. For instance, when dealing with fractions, you can quickly simplify them by checking for common factors using divisibility rules. In problems involving large numbers, applying these rules can save significant time and reduce the chance of calculation errors.
Consider a problem asking for the remainder when a large number is divided by a small number. Often, you can use divisibility rules to simplify the number or identify patterns. For example, if you need to find the remainder of a large number when divided by 9, you simply find the remainder of the sum of its digits when divided by 9.
Practice and Mastery
Consistent practice is key to mastering divisibility rules. Work through a variety of problems, starting with simple checks and progressing to complex applications. The more you use these rules, the more intuitive they will become, allowing you to solve quantitative aptitude questions efficiently and accurately.
Learning Resources
Provides a clear and concise explanation of common divisibility rules with examples, suitable for exam preparation.
A comprehensive guide specifically tailored for competitive exams, covering rules and their applications.
Offers a structured approach to divisibility rules with practice questions and explanations.
Khan Academy video explaining basic divisibility tests for 2, 3, and 5, with visual aids.
Continues the series from Khan Academy, covering divisibility tests for more numbers.
A community forum for mathematical questions, where you can find discussions and explanations on divisibility concepts.
Explains the concept of divisibility and its fundamental properties in an easy-to-understand manner.
A resource page for CAT quantitative aptitude, often including sections on number systems and divisibility.
Offers a collection of practice questions focused on divisibility rules to test your understanding.
Provides a broader overview of number theory, including concepts related to divisibility and prime numbers.