I'm working on a college assignment where we were given a system and asked to exploit it. I was given a Fedora 13 Goddard running Linux kernel 126.96.36.199, and my task is to gain root privileges on the system and report the vulnerability which was exploited. We are free to borrow online help too. The problem is that I have been searching for a while, but couldn't find any exploit that can grant root privileges. I tried various exploits from packet storm, exploit DB and others, the best I can get is a DoS. I've seen the CVE databases, and I've found few vulnerabilities detailed, but can't find any exploit code anywhere. I'd be really glad if someone can help me out with this. Thanks in advance.
Look for when the last patch was applied, and then look for Linux privilege escalation vulnerabilities that are not patched. Most of the Linux privilege escalation vulnerabilities affect multiple distros.
In this case, Linux kernel 188.8.131.52 seems very old, so public exploits such as "dirty cow" (CVE-2016-5195) may work.
Other methods may involve brute forcing the admin password.