In-Depth: LCM and HCF

CLAT Application & Relevance

Importance: Low. While not directly tested as standalone complex problems, LCM (Least Common Multiple) and HCF (Highest Common Factor) are foundational concepts. They can be indirectly useful for simplifying fractions/ratios in Data Interpretation, or in specific problem types like "Time and Work" (using LCM to determine total work) or problems involving meeting points/coincidences.

How it's tested: Implicitly in simplification steps, or as a very basic calculation within a larger problem, often involving scenarios that require finding a common cycle or distribution.

Section 1: Core Concepts & Methods

HCF (Highest Common Factor) / GCD (Greatest Common Divisor): The largest common divisor of two or more given numbers.

LCM (Least Common Multiple): The smallest positive integer that is divisible by all the given numbers.

Methods to Find HCF & LCM

1. Prime Factorization Method:

Step 1: Find the prime factorization of each number.
Step 2 (for HCF): Multiply the common prime factors (each to its lowest power).
Step 3 (for LCM): Multiply all prime factors (each to its highest power).

2. Division Method (for HCF): Successively divide the larger number by the smaller number until the remainder is zero. The last divisor is the HCF.

3. Division Method (for LCM): Divide the numbers by common prime factors until no common factor remains. Multiply all the divisors and the remaining undivided numbers.

Key Formulas & Properties

For two numbers 'a' and 'b':
Product of two numbers = HCF(a, b) × LCM(a, b)
a × b = HCF × LCM
HCF of fractions: HCF of Numerators / LCM of Denominators
LCM of fractions: LCM of Numerators / HCF of Denominators

Section 2: Solved CLAT-Style Examples (Application)

Example 1: Application in Time-Based Problems (Caselet)

Passage Context: "Three lawyers, A, B, and C, are preparing for a series of legal webinars. Lawyer A prepares a new topic every 4 days. Lawyer B prepares a new topic every 6 days. Lawyer C prepares a new topic every 8 days. If they all launched a new topic today, and they want to launch a combined special topic on the next day they all happen to launch a new topic simultaneously."

Question: "After how many days will they next launch a new topic simultaneously?"

Detailed Solution:
1. Understand the problem: We need to find the smallest number of days that is a multiple of 4, 6, and 8. This is the LCM.
2. Find the LCM of 4, 6, and 8: - Prime factorization: 4 = 2² 6 = 2 × 3 8 = 2³ - LCM = Product of highest powers of all prime factors = 2³ × 3¹ = 8 × 3 = 24.
Answer: They will next launch a new topic simultaneously after 24 days.

Example 2: Using HCF for Distribution Problems (Caselet)

Passage Context: "A legal aid society has 144 pro bono cases involving civil disputes and 108 pro bono cases involving criminal matters. They want to form an equal number of teams, with each team having the same number of civil cases and the same number of criminal cases. All cases must be distributed among these teams."

Question: "What is the maximum number of such teams that can be formed?"

Detailed Solution:
1. Understand the problem: We need to find the largest number that divides both 144 and 108 evenly. This is the HCF.
2. Find the HCF of 144 and 108 (using Prime Factorization): - Prime factorization: 144 = 2⁴ × 3² 108 = 2² × 3³ - HCF = Product of common prime factors to their lowest power = 2² × 3² = 4 × 9 = 36.
3. Verify: If 36 teams are formed: Civil cases per team = 144 / 36 = 4 cases. Criminal cases per team = 108 / 36 = 3 cases. This forms teams with identical composition.
Answer: A maximum of 36 such teams can be formed.