$\mathrm{ord}_{p}(a+b)\ge\mathrm{min}(\mathrm{ord}_{p}a,\mathrm{ord}_{p}b)$ with equality holding if $\mathrm{ord}_{p}a\ne \mathrm{ord}_{p}b$. is a the statement that prompted this question.
It was found in Ireland & Rosen's Elements of Number Theory (precurser to their book A Classical Introduction to Modern Number Theory) is the book that I am working through and it asks for its proof.
After some research I'm learned that this function is completely additive($\mathrm{ord}_{p}(ab)=\mathrm{ord}_{p}(a)+\mathrm{ord}_{p}(b)$) among other things and managed to get the following equation out that if I didn't make a mistake, solves the first part of the question:
$\mathrm{ord}_{p}(a+b)=\mathrm{ord}_{p}(dm+dn)=\mathrm{ord}_{p}(d)+\mathrm{ord}_{p}(m+n)\ge \mathrm{ord}_{p}(d)=\mathrm{ord}_{p}(a,b)=\mathrm{min}(\mathrm{ord}_{p}a,\mathrm{ord}_{p}b)$
Where $(a,b)$=$\mathrm{gcd}(a,b)$ is the ideal/greatest common factor and is equal to $d$, and $a=dm$, $b=dn$ where $m$,$n$ are relatively prime.
So my question is where can I learn more about the properties and identities of the functions $\mathrm{ord}_{p}(n)$ and $\mathrm{v}(n)$ (the latter seems to be called valuation or related to such), especially a resource that would include relations similar to those above. If the resource included identities for the lcm, gcd, and min/max functions in relation to the ord function and by themeslves also, that would be wonderful.
Also what are some identities that could solve the "with equality holding if $\mathrm{ord}_{p}a\ne \mathrm{ord}_{p}b$." part and how are they derived? Edit(This part has been answered, looking for some good resources that expound on ord,v, and related functions.)
Thank you for your assistance.