LCM and HCF

Formula / Rule

LCM and HCF

Topic

Number System

Formula

Relationship between two numbers (A, B): $LCM(A, B) \times HCF(A, B) = A \times B$

Definitions:

LCM (Least Common Multiple): The smallest positive integer that is divisible by both given numbers.
HCF (Highest Common Factor): The largest positive integer that divides both given numbers (also known as GCD - Greatest Common Divisor).

Finding Methods (using prime factorization):

HCF: Take the smallest powers of all common prime factors.
LCM: Take the highest powers of all prime factors (common and uncommon).

When to Use

Finding smallest/largest common values, solving interval/scheduling problems.

Worked Example

Q: Find LCM and HCF of 12 and 18.

$12 = 2^2 \times 3$, $18 = 2 \times 3^2$ $HCF = 2^1 \times 3^1 = \textbf{6}$ $LCM = 2^2 \times 3^2 = \textbf{36}$ Check: $6 \times 36 = 216 = 12 \times 18 \checkmark$