lightbulb Key Concepts

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1. Euclid's Division Lemma

Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, where 0 ≤ r < b.

2. Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this factorization is unique.

• HCF: Product of the smallest power of each common prime factor.
• LCM: Product of the greatest power of each prime factor.

3. Irrational Numbers

A number is irrational if it cannot be written in the form p/q (e.g., √2, √3, π). Theorem: If p is prime and p divides a², then p divides a.