LCM and HCF

Overview & Fundamental Concept

Pointwise AISSEE Class 6 Mathematics guide for LCM and HCF (Topic 2). Master prime factorization, division method, co-prime rules, golden product formula, and word problems on bells and maximum containers.

• Factors: A factor of a number is an exact divisor that leaves zero remainder. The number 1 is a factor of every number. Every number is a factor of itself. The number of factors of any finite natural number is always finite.
• Multiples: A multiple of a number is the product obtained by multiplying that number by any natural number. Every number is a multiple of itself. The number of multiples of any number is infinite.
• Prime Factorisation: Expressing a composite number strictly as a product of prime numbers. For example: $72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$. Every composite number has a unique prime factorisation (Fundamental Theorem of Arithmetic).
• Co-Prime Numbers: Two numbers are said to be co-prime (or mutually prime) if their Highest Common Factor is 1 (e.g., $8$ and $15$, or any two consecutive natural numbers like $14$ and $15$). Co-prime numbers do NOT need to be prime numbers individually.

• Definition of HCF: The Highest Common Factor (also called Greatest Common Divisor or GCD) of two or more numbers is the largest number that divides each of the given numbers exactly without leaving any remainder.
• Prime Factorisation Method for HCF: 1. Find the prime factorisation of each number. 2. Identify the common prime factors. 3. Take the lowest power (exponent) of each common prime factor. 4. Multiply them.
• Division Method (Euclidean Method): Divide the larger number by the smaller number. If remainder $\ne 0$, divide the previous divisor by this remainder. Repeat until the remainder becomes 0. The last divisor is the HCF.
• HCF of Fractions: $\text{HCF of Fractions} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$.

• Definition of LCM: The Lowest Common Multiple of two or more numbers is the smallest positive number that is a multiple of all the given numbers.
• Prime Factorisation Method for LCM: 1. Write the prime factorisation of each number. 2. Take the highest power (exponent) of EVERY prime factor that appears in any of the factorisations. 3. Multiply them together.
• Common Division (Ladder) Method: Write the numbers in a row separated by commas. Divide by the smallest prime number that divides at least two of the numbers. Carry down undivided numbers. Continue until no two numbers share a common prime factor. LCM is the product of all divisors and remaining quotients.
• LCM of Fractions: $\text{LCM of Fractions} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$.

• The Fundamental Product Identity: For any two positive integers $a$ and $b$: $$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$$ Hence: $a = \frac{\text{HCF} \times \text{LCM}}{b}$ and $\text{LCM} = \frac{a \times b}{\text{HCF}}$. Caution: This formula applies ONLY to pairs of two numbers, never to three numbers.
• HCF Word Problem Signatures: Keywords like 'greatest length', 'maximum capacity', 'largest measuring container', or 'dividing items equally without remainder' require finding the HCF.
• LCM Word Problem Signatures: Scenarios involving cyclical events that repeat at regular intervals: 'bells tolling together', 'traffic lights changing simultaneously', or 'runners completing laps on a track and meeting at the start' require finding the LCM.
• Remainder Models:
  • Smallest number divided by $x, y, z$ leaving remainder $r$ in each case $= \text{LCM}(x, y, z) + r$.
  • Greatest number that divides $x, y, z$ leaving remainder $r$ in each case $= \text{HCF}(x - r, y - r, z - r)$.