Significance of GCD and Coprimality of Integers

The greatest common factor (GCF) or greatest common divisor (GCD) of a set of integers is the largest positive integer that divides all of the integers. The GCF of a set of integers $a_1, a_2, ..., a_n$ is typically denoted $(a_1, a_2, ..., a_n)$.

Here are some examples: $$(12,18)=6$$ $$(2,4)=2$$ $$(-75,120)=15$$ $$(0,126,49)=7$$

Multiple numbers are said to be relatively prime if their GCF equals $1$.

Here are some examples of sets of relatively prime integers: $$1,2,3,4,5$$ $$6,10,15$$ $$16087,16088$$

Suppose that we have two numbers, and we take their prime factorizations, as follows: $$A=\prod_{i=1}^{k}(p_i)^{\alpha_i}$$ $$B=\prod_{i=1}^{k}(p_i)^{\beta_i}$$ Then, it can be easily shown that $$(A,B)=\prod_{i=1}^{k}(p_i)^{\min(\alpha_i, \beta_i)}.$$