HCF & LCM

Fundamental Theorem of Arithmetic: Every integer greater than 1 is either a prime number or can be uniquely represented as a product of prime numbers. This 'prime factorization' is the foundation for calculating HCF and LCM systematically.

The Product Rule: For any two positive integers $a$ and $b$, the product of their HCF and LCM is equal to the product of the numbers themselves. This is expressed by the formula: $HCF(a, b) \times LCM(a, b) = a \times b$

Divisibility Logic: If a number $d$ is the HCF of $a$ and $b$, then $a/d$ and $b/d$ must be coprime (meaning they share no common factors other than 1).

Check for Divisibility: If the larger number is exactly divisible by the smaller number, the smaller number is the HCF and the larger number is the LCM. For example, with 5 and 20, HCF is 5 and LCM is 20.

Coprime Shortcut: If two numbers are prime or share no common factors (coprime), their HCF is always 1 and their LCM is simply their product ($a \times b$).

Sanity Check: Always verify that your HCF divides into the original numbers and that the original numbers divide into your LCM. If they don't, a calculation error has occurred in the prime factorization or multiplication steps.

Confusing Names: Students often assume 'Highest' means the result will be a large number and 'Lowest' means a small number. In reality, the HCF is a small divisor and the LCM is a large multiple.

Missing Prime Factors: When using the prime power method for LCM, students often forget to include prime factors that appear in only one of the numbers. LCM must include the highest power of every prime factor present in the set.

Intersection Errors: In Venn diagrams, if a prime factor appears multiple times (e.g., $2^3$), ensure the correct number of instances are placed in the intersection based on the minimum count shared by both numbers.