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Number System & Divisibility Rules - Aptitude Mastery Guide

Section titled “Number System & Divisibility Rules - Aptitude Mastery Guide”

Category: Quantitative Aptitude
Generated on: 2025-07-15 09:14:47
Source: Aptitude Mastery Guide Generator

Number System & Divisibility Rules: A Comprehensive Guide

Section titled “Number System & Divisibility Rules: A Comprehensive Guide”

This guide is your ultimate resource for mastering Number Systems and Divisibility Rules, crucial for success in quantitative aptitude sections of competitive exams and placement tests. We’ll cover foundational concepts, powerful shortcuts, essential formulas, solved examples, and practice problems. Prepare to elevate your problem-solving skills!

1. Foundational Concepts

Section titled “1. Foundational Concepts”

The Number System is the foundation of arithmetic. Understanding different types of numbers and their properties is essential.

  • Natural Numbers (N): Positive integers starting from 1 (1, 2, 3, …).
  • Whole Numbers (W): Natural numbers including 0 (0, 1, 2, 3, …).
  • Integers (Z): All positive and negative whole numbers, including 0 (… -3, -2, -1, 0, 1, 2, 3 …).
  • Rational Numbers (Q): Numbers that can be expressed as a fraction p/q, where p and q are integers, and q ≠ 0 (e.g., 1/2, -3/4, 5).
  • Irrational Numbers: Numbers that cannot be expressed as a fraction p/q (e.g., √2, π). These have non-repeating, non-terminating decimal expansions.
  • Real Numbers (R): The set of all rational and irrational numbers.
  • Prime Numbers: Numbers greater than 1 that have only two factors: 1 and themselves (e.g., 2, 3, 5, 7, 11, 13…). 2 is the only even prime number.
  • Composite Numbers: Numbers greater than 1 that have more than two factors (e.g., 4, 6, 8, 9, 10…).
  • Co-prime Numbers (Relatively Prime): Two numbers are co-prime if their greatest common divisor (GCD) is 1 (e.g., 8 and 15). They don’t necessarily have to be prime themselves.
  • Even Numbers: Numbers divisible by 2. Can be represented as 2n, where n is an integer.
  • Odd Numbers: Numbers not divisible by 2. Can be represented as 2n+1, where n is an integer.

Understanding Place Value:

In the decimal system (base-10), each digit’s position represents a power of 10. For example, in the number 567, 5 is in the hundreds place (10^2), 6 is in the tens place (10^1), and 7 is in the ones place (10^0). This is crucial for understanding number manipulation.

Why Place Value Matters: It allows us to easily break down numbers and perform arithmetic operations. For instance, 567 = (5 * 100) + (6 * 10) + (7 * 1).

2. Key Tricks & Shortcuts

Section titled “2. Key Tricks & Shortcuts”

This section focuses on practical tricks and shortcuts to significantly improve your speed and accuracy.

  • Divisibility Rules: These are the cornerstone of simplifying calculations. (See the table in Section 3).

  • Finding the Last Digit of a Power (Cyclicity):

    • Concept: The last digit of a number raised to different powers often follows a cyclical pattern.
    • How to Use: Find the cycle of the last digit for the base number. Divide the power by the length of the cycle. The remainder determines the last digit.
    • Example: Find the last digit of 725. The last digits of powers of 7 are: 71 = 7, 72 = 49 (9), 73 = 343 (3), 74 = 2401 (1). The cycle is 7-9-3-1 (length 4). 25 divided by 4 leaves a remainder of 1. Therefore, the last digit of 725 is 7 (the first digit in the cycle).

  • Vedic Maths - Base Method for Multiplication (Faster Multiplication):

    • Concept: This method simplifies multiplication when both numbers are close to a common base (e.g., 10, 100, 1000).
    • How to Use:
      1. Choose a base close to both numbers.
      2. Find the difference (deviation) of each number from the base.
      3. Add the deviation of one number to the other number. This gives you the first part of the answer.
      4. Multiply the deviations. This gives you the second part of the answer.
      5. Combine the two parts, adjusting for the base.
    • Example: Multiply 104 x 107.
      1. Base = 100
      2. Deviations: 104 is +4 from the base, and 107 is +7 from the base.
      3. (104 + 7) or (107 + 4) = 111 (First part)
      4. 4 x 7 = 28 (Second part)
      5. Answer: 11128

  • Vedic Maths - Squaring Numbers Ending in 5:

    • Concept: A quick way to square numbers ending in 5.
    • How to Use:
      1. Take the digits before the 5.
      2. Multiply those digits by the next higher integer.
      3. Append 25 to the result.
    • Example: Square 65.
      1. Digits before 5: 6
      2. 6 x 7 = 42
      3. Append 25: 4225. Therefore, 652 = 4225.

  • Percentage-to-Fraction Conversions:

    • Concept: Knowing common percentage-to-fraction equivalents allows for faster calculations in percentage-related problems.
    • Examples: 50% = 1/2, 25% = 1/4, 20% = 1/5, 10% = 1/10, 33.33% (or 33 1/3%) = 1/3, 16.66% (or 16 2/3%) = 1/6, 12.5% = 1/8, 6.25% = 1/16.

  • Assumption Method (for Remainder Problems):

    • Concept: When dealing with remainders, assume a suitable number that satisfies the given conditions. This often simplifies the problem.
    • How to Use: Read the problem carefully to identify the conditions. Assume a number that meets those conditions. Perform the required operations on the assumed number. Adjust the result to find the actual remainder.
    • Example: A number when divided by 7 leaves a remainder of 3. What is the remainder when the square of the number is divided by 7?
      • Assume the number is 10 (7 + 3).
      • The square of the number is 100.
      • When 100 is divided by 7, the remainder is 2.

  • Sum of First ‘n’ Natural Numbers, Squares, and Cubes (Useful for Series Problems): (These are in the formulas section, but worth highlighting here).

3. Essential Formulas & Rules

Section titled “3. Essential Formulas & Rules”
Rule/FormulaDescription
Divisibility by 2The last digit is even (0, 2, 4, 6, 8).
Divisibility by 3The sum of the digits is divisible by 3.
Divisibility by 4The number formed by the last two digits is divisible by 4.
Divisibility by 5The last digit is 0 or 5.
Divisibility by 6The number is divisible by both 2 and 3.
Divisibility by 7Double the last digit and subtract it from the remaining digits. If the result is divisible by 7, then the original number is divisible by 7. Repeat if needed.
Divisibility by 8The number formed by the last three digits is divisible by 8.
Divisibility by 9The sum of the digits is divisible by 9.
Divisibility by 10The last digit is 0.
Divisibility by 11The difference between the sum of digits at odd places and the sum of digits at even places is either 0 or divisible by 11.
Divisibility by 12The number is divisible by both 3 and 4.
Sum of first ‘n’ natural numbersn(n+1)/2
Sum of squares of first ‘n’ natural numbersn(n+1)(2n+1)/6
Sum of cubes of first ‘n’ natural numbers[n(n+1)/2]2
an - bn is divisible by(a - b) for all values of n, and by (a + b) when n is even.
an + bn is divisible by(a + b) when n is odd.
Number of factors of N = paqbrc…(a+1)(b+1)(c+1)… where p, q, r are prime factors.
Sum of factors of N = paqbrc…[(p0 + p1 + … + pa) * (q0 + q1 + … + qb) * (r0 + r1 + … + rc) …]
LCM * HCF = Product of two numbersApplies only to two numbers.

4. Detailed Solved Examples

Section titled “4. Detailed Solved Examples”

Example 1: Basic Divisibility Rule [Easy]

  • Problem: Is the number 345678 divisible by 6?
  • Solution:
    1. Divisibility by 2: The last digit is 8, which is even. So, the number is divisible by 2.
    2. Divisibility by 3: The sum of the digits is 3 + 4 + 5 + 6 + 7 + 8 = 33. 33 is divisible by 3. So, the number is divisible by 3.
    3. Conclusion: Since 345678 is divisible by both 2 and 3, it is divisible by 6.

Example 2: Finding the Remainder [Medium]

  • Problem: What is the remainder when 123456789 is divided by 9?
  • Solution:
    1. Divisibility Rule for 9: The sum of the digits is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.
    2. Divisibility Check: 45 is divisible by 9 (45 / 9 = 5).
    3. Conclusion: Therefore, the remainder when 123456789 is divided by 9 is 0.

Example 3: Cyclicity of Last Digits [Medium]

  • Problem: Find the last digit of 352.
  • Solution:
    1. Find the Cycle:
      • 31 = 3
      • 32 = 9
      • 33 = 27 (7)
      • 34 = 81 (1)
      • 35 = 243 (3) The cycle of the last digits is 3-9-7-1 (length 4).
    2. Divide the Power: 52 / 4 = 13 (remainder 0). A remainder of 0 means we take the last digit of the cycle.
    3. Conclusion: The last digit of 352 is 1.

Example 4: Applying Vedic Maths (Base Method) [Medium]

  • Problem: Calculate 97 * 96 using the Base Method.
  • Solution:
    1. Choose a Base: The base is 100.
    2. Find Deviations: 97 is -3 from the base, 96 is -4 from the base.
    3. Cross-Add: (97 - 4) or (96 - 3) = 93 (First Part)
    4. Multiply Deviations: (-3) * (-4) = 12 (Second Part)
    5. Combine: The answer is 9312.

Example 5: Number of Factors [Hard]

  • Problem: Find the number of factors of 360.
  • Solution:
    1. Prime Factorization: 360 = 23 * 32 * 51
    2. Apply the Formula: Number of factors = (3+1)(2+1)(1+1) = 4 * 3 * 2 = 24
    3. Conclusion: 360 has 24 factors.

5. Practice Problems (Graded Difficulty)

Section titled “5. Practice Problems (Graded Difficulty)”

[Easy] 1. Is 12345 divisible by 5? [Easy] 2. Is 729 divisible by 9? [Medium] 3. What is the remainder when 535 is divided by 12? [Medium] 4. Find the last digit of 8123. [Medium] 5. How many factors does 100 have? [Hard] 6. Find the sum of all factors of 48. [Hard] 7. A number when divided by 5 leaves a remainder of 2, and when divided by 7 leaves a remainder of 3. What is the smallest such number?

6. Advanced/Case-Based Question

Section titled “6. Advanced/Case-Based Question”

A fruit vendor has a certain number of apples. He arranges them in rows of 6, 8, or 12, and each time he is left with 5 apples. He then tries arranging them in rows of 11 and finds that there are no apples left. What is the minimum number of apples he could have? (Hint: This problem combines LCM and divisibility rules).