Prime Factorization

A factor tree is the primary and arguably most simple way to perform prime factorization. It involves finding a factor pair of a number, then writing those numbers as "brances" from the main number. You continue to split each of the branches into more and more branches until you end at a prime number. Typically, best practice is to divide by smaller values at first, such as \(2\) if you know it's even, in order to ensure that the prime factorization remains on one or two key branches

The prime factorization of a number, written as \(n=a^x \times b^y...\) for prime numbers \(a,b,c\) can also tell you the number of factors of \(n\) as a whole. To do this, raise each exponent by \(1\) and multiply them together. For example, if we know the prime factorization of \(60=2^2 \times 3 \times 5\), we would find \((2+1) \times (1+1) \times (1+1)=12\). This works because for each factor, each prime factor can have up to \(n\) repetitions in the prime factorization of the factor, or none, which is why we add \(1\)

GCF

The \(\textup{GCF}\), also known as \(\textup{GCD}\) is the "Greatest Common Factor/Denominator" of a set of one or two numbers. Effectively it is the largest integer that divides two numbers. Using the prime factorization, you can find the \(\textup{GCF}\) of two numbers by finding both of their prime factorizations and multiplying all of their common prime factors. Obviously, these factors, as well as their product, is a factor of a number, and since they are in the prime factorizations of both, they must be the prime factors of both. For example, \(60=2^2 \times 3 \times 5\) and \(45=3^2 \times 5\) have a \(\textup{GCF}\) of \(15\) because their overlapping factors are \(3 \times 5\). Knowing the \(\textup{GCF}\) is a valuable tool for cross multiplication. For example, if you have \(\frac{1}{4} \times \frac{2}{3}\), you would take the \(\textup{GCF}\) of \(2\) and \(4\), and factor it out. This allows you to save time and understand number theory further

LCM

The \(\textup{LCM}\) of two numbers is the smallest integer that they are both factors of. It is heavily related to the \(\textup{GCF}\) because if you know the \(\textup{GCF}\) of two numbers, then proceed to divide both of the numbers by the \(\textup{GCF}\), and multiply each quotient by the OTHER number, you get the LCM. This makes sense because it's effectively factoring out the GCF. Notice that this also means that the \(\textup{LCM}\) of coprime numbers (numbers who share no factors other than \(1\) will be the product of both of them combined).