I could solve this problem using a brute force mentality but I am looking for an elegant logical solution. So far I have tried to narrow down the possibilities and have found that:
$x > 22$ as any number, $c < 22$ when divided by $22$ has a quotient $= 0$ and a remainder $\not=$ to the quotient.
$x \thinspace\%\thinspace 22 \not=0$ so $x$ must be odd for remainder $\not= 0$.
What is the largest number, $x$, which when divided by $22$ has a quotient equal to the remainder?