What is the largest $4$ digit integer congruent to $15 \pmod{22}?$
I need to find a number that has a remainder of $15$ when divided by $22$. I can start by finding the largest multiple of $22$ that is less than $10000$, since that would be the largest $4$ digit number. To do that, I can divide $10000$ by $22$ and round down to get the quotient. I get $10000 \div 22 \approx 454$, so $454 \times 22 = 9988$ is the largest multiple of $22$ that is less than $10000$. Now, to get the largest number that is congruent to $15 \pmod{22}$, I just need to add $15$ to $9988$. That gives me $9988 + 15 = 10003$. But wait, that is not a $4$ digit number, it is a $5$ digit number. So, I need to go down one more multiple of $22$ and add $15$ to that. The next smaller multiple of $22$ is $9988 - 22 = 9966$. This is a $4$ digit number, and it is the largest one that is congruent to $15 \pmod{22}$. 