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Cheatsheets - General

  • LCM - The LCM (Lowest Common Multiple) is the product of the highest powers of all prime factors
  • GCF - The GCF (Greatest Common Factor) is the product of the lowest powers of common prime factors.
  • The relationship between the Least Common Multiple (LCM) and the Greatest Common Factor (GCF) of two numbers \( a \) and \( b \) can be expressed by the following formula:
\[ \text{LCM}(a, b) \times \text{GCF}(a, b) = |a \times b| \]

thus the LCM is:

\[ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCF}(a, b)} \]

and the GCF is:

\[ \text{GCF}(a, b) = \frac{|a \times b|}{\text{LCM}(a, b)} \]

  • Prime:

    • for \(n\) > 1, 2 distinct factors: 1 and itself

    • building blocks of all natural numbers > 1

    • 2 is the smallest and the only even prime number since \(2n\) are no longer primes.

  • Coprime or Relative Prime:

    • when \(\text{GCF}(a, b) = 1\); In other words, no common factors other than 1.

  • Fundamental Theorem of Arithmetic: Every natural number greater > 1 can be uniquely expressed as a product of prime numbers, disregarding the order of the factors.

  • Fundamental Rule of Algebra - states that every non-constant polynomial function with complex coefficients has at least one complex root. Moreover, a polynomial of degree \(n\) will have exactly \(n\) roots in the complex number system, counting multiplicities.