Suppose I have a positive integer $a$ which can be written as $\equiv n (mod) t$ for some $n$ and $t$. Now if I have another integer $b$ related to $a$ as: $b=\sqrt{(2t+1)a^2+t^2}$, and I can write $b \equiv o (mod) t$; then does there exist a simple relation between $n$ and $o$? If yes, how can I proceed to find it? More specifically, given $n$ and $t$, can $o$ be found? The question arises since I do not want to deal with potentially very large squares if $b$ is large.
For example, I can write $a=160$ as $\equiv 10(mod)30$. $b$ is then $1250$ which can be written as $\equiv 20(mod)30$. The question is to have a simple relation between $n=10$ and $o=20$.