LCM and HCF

Overview

LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number theory at the primary level. These concepts appear directly in OTET Paper I Mathematics and also serve as building blocks for fractions, ratio-proportion, and word problems involving time, work, and distribution.

For OTET, you must master three things: (1) the definitions and distinction between LCM and HCF, (2) multiple methods to calculate them, and (3) their application in real-life word problems. Questions typically test your speed in finding LCM/HCF using prime factorisation or division method, and your ability to identify which concept applies in a given situation.

Understanding the relationship between LCM and HCF is equally important, as questions often involve finding one when the other is given, along with the product of two numbers.

Key Concepts

• **Factor**: A number that divides another number exactly (without remainder). Factors of 12: 1, 2, 3, 4, 6, 12.

• **Multiple**: A number obtained by multiplying a given number by any whole number. Multiples of 4: 4, 8, 12, 16, 20...

• **HCF (Highest Common Factor)**: The greatest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor). HCF is always smaller than or equal to the smallest given number.

• **LCM (Lowest Common Multiple)**: The smallest number that is a multiple of two or more numbers. LCM is always greater than or equal to the largest given number.

• **Co-prime numbers**: Two numbers whose HCF is 1 (e.g., 8 and 15). For co-primes, LCM = product of the numbers.

• **Relationship formula**: For any two numbers a and b: LCM × HCF = a × b. This is a frequently tested concept.

• **When to use HCF**: Problems involving division, distribution into equal groups, cutting/measuring with maximum size.

• **When to use LCM**: Problems involving repetition, cycles, finding when events coincide again.

Formulas / Key Facts

| Concept | Formula/Fact | |---------|--------------| | Fundamental relationship | LCM(a, b) × HCF(a, b) = a × b | | Finding one from other | LCM = (a × b) ÷ HCF | | HCF of co-primes | HCF = 1 | | LCM of co-primes | LCM = Product of the numbers | | HCF of consecutive numbers | Always 1 | | LCM of consecutive numbers | Product of the numbers | | HCF ≤ Smallest number | Always true | | LCM ≥ Largest number | Always true | | HCF divides LCM | Always true |

**Prime Factorisation Method:**

• HCF = Product of common prime factors with lowest powers