Whilst reading the solutions to a question I encountered the symbol "|" (vertical line) in the format of a|b (the exact replica of what was written is "q|q^2"). After googling, the only definition which I have found is "or", which does not make sense in context.
Context: Question： Let p and q respectively be the smallest and largest prime factors of n. Find all positive integers n such that p^2 + q^2 = n + 9
Solution given: First eliminate the special case where n = p = q , which leads to 2p^2 = p + 9. This has no integer solutions. Rewrite the equation as p^2+q^2−9=kpq for some positive integer k Now, since q|q^2 and q|kpq, we have q|p^2 − 9 = (p − 3)(p + 3) and, since q is prime, q|p−3 or q|p+3 Hence either p=3 or q|p+3 If p=3 then n=q^2 and 3|n so q=3 and n=9, which works. If q|p+3 and p=2 then q=5 and n=20 , which works. If p>3 then q> \frac {p+3}{2} and so q=p+3, which is even. So this is impossible. Hence only solutions are n=9 and n=20.
What does | mean in this context? And just out of curiosity, what are its meanings in other contextes?
Apologies for any errors in formatting. I'm still trying to get the gist of MathJax. And apologies too for any errors in tagging--I found this symbol under the "Logic" section when conducting my unfruitful google search, so that's how I tagged it.
If the formatting is so abysmal as to be completely unreadable, the question & solution can be found at ukmt.org, under Intermediate, Maclaurin Olympiad, past papers, 2020.
Thanks for the help.