What is the largest negative integer $x$ satisfying $24x \equiv 15 \pmod{1199}?$ What is the largest negative integer $x$ satisfying $$24x \equiv 15 \pmod{1199}~?$$
I do not know how to simplify this, any help?  Thanks for taking the time to answer my question.

You have to first solve this congruence, finding the inverse of $24$ modulo $1199$, which is easy: $\;50\times 24=1200\equiv 1\mod1199$. So multiply both sides of the congruence by $50$. This yields
$$50\times 24 x\equiv x\equiv 50\times 15=750\equiv -449\mod 1199.$$
