Overview

  • Divisibility rules help quickly decide if a number can be divided by another number without performing full division.
  • They are crucial tools for solving factorization, LCM, HCF, number system, and simplification problems.
  • In competitive exams, using these rules saves time and reduces calculation errors.

Applications in Factorization

  • Identify prime factors of a number efficiently.
  • Example: Check if 252 is divisible by 2 (even), 3 (sum=9), and so on.
  • Helps in breaking numbers into smaller components.

Applications in Finding HCF and LCM

  • Determine common factors quickly by applying divisibility tests to each number.
  • Example: To find HCF of 72 and 108, note that both are divisible by 2, 3, 4, 6, 9.
  • LCM requires identifying divisibility to ensure no factors are left out.

Applications in Simplification

  • Helps to reduce fractions to the simplest form.
  • Example: Simplify 84/126 by checking divisibility by 2, 3, 7.
  • Speeds up simplification of algebraic expressions involving numeric coefficients.

Applications in Remainder Problems

  • Divisibility rules assist in quickly predicting if a number will leave a remainder.
  • Example: If sum of digits is not divisible by 9, the remainder will not be zero when dividing by 9.
  • Essential in modular arithmetic problems.

Applications in Checking Multiples

  • Quickly confirm if a number is a multiple of 2, 3, 5, or 10.
  • Useful for verifying answers in series and sequence questions.
  • Example: In series problems, confirm each term’s divisibility by the required base.

Applications in Number Series

  • Divisibility tests help detect patterns in series.
  • Example: Identify terms divisible by 11 or 7 in a given sequence.
  • Speeds up solving missing term or wrong term questions.

Applications in Divisibility-based Word Problems

  • Questions involving minimum numbers to be added or subtracted to make a number divisible by a given divisor.
  • Example: What is the smallest number to add to 43 to make it divisible by 5?
  • Approach: Check nearest multiple of 5 above 43 (45), and subtract 43.

Applications in Finding Common Multiples

  • Identify smallest common multiple using divisibility rules.
  • For example, LCM of 8 and 12: Find smallest number divisible by both using rules for 2 and 4.

Examples of Divisibility Application

  • Divisibility by 2: Check if the last digit is even.
  • Divisibility by 3: Sum digits and check if divisible by 3.
  • Divisibility by 4: Last two digits divisible by 4.
  • Divisibility by 5: Last digit 0 or 5.
  • Divisibility by 6: Divisible by both 2 and 3.
  • Divisibility by 7: Double last digit, subtract from rest.
  • Divisibility by 8: Last three digits divisible by 8.
  • Divisibility by 9: Sum digits divisible by 9.
  • Divisibility by 10: Last digit 0.
  • Divisibility by 11: Difference of sums of alternate digits divisible by 11.

Common Mistakes to Avoid

  • Forgetting to check all rules in combined cases (e.g., divisibility by 6).
  • Confusing rules for 4 and 8.
  • Misapplying alternate digit sum rule for 11.

Benefits of Mastery

  • Boosts speed and accuracy.
  • Helps in mental math calculations.
  • Provides confidence in number system, arithmetic, and reasoning questions.

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