How many three-digit positive integers $x$ satisfy $3874x+481\equiv 1205 \pmod{23}$?
Let's compute $1205-481$ first. That's $724$. So we are looking for $x$ such that $3874x\equiv 724 \pmod{23}$ So we have $3874x\equiv 11\pmod{23}$. Yes, so we have $10x\equiv 11\pmod{23}$. Yes, so we just need to find the last three-digit number that satisfies this. So $23k<900$. Right. So there are $40$ three-digit positive integers $x$ that satisfy $3874x+481\equiv 1205 \pmod{23}$.

