Let me express your question a bit more constructively:
Question Rephrased:
If you take a number $a$ and split it into it's upper bits and lower bits at bit $n$ for an $N$ bit number:
$a = 2^{(N-n)} + r$
where $r$ is the remainder, and $N \ge n$. Then you can easily see that:
$a \equiv 2^{(N-n)} \pmod r$
for your example $5 = 2^2+1$ when split at bit number 2, then $5 \equiv 2^2 \pmod 1$ and 11 is $2^3+3$ when split at bit 3, then $11 \equiv 2^3 \pmod 3$.
So the question you are asking is, is it possible to figure out something about a composite number $c = a_1 \times a_2$ by looking at its factors and examining to see if something can be derived examining $r_1$ and $r_2$ by noting that $a_1 \equiv 2^{k_1} \pmod {r_1}$ and $a_2 \equiv 2^{k_2} \pmod {r_2}$ for some ${k_1}$ and ${k_2}$, where ${k_1}$ and ${k_2}$ are as yet unknown.
answer:
The answer in general is no you cannot derive interesting information, because your choice of where to split the number (bits $n_1$ and $n_2$) is not known.
$k_1 = N-n_1$ and $k_2 = N-n_2$ and you stipulated that you don't know $k_1$ and $k_2$.
Now the next question is, what if you randomly chose $n_1$ and $n_2$? Well you are now starting to stumble upon probabilistic methods of tests of primality.
You still need to put strong limits on choices of $n_1$ and $n_2$. As an exercise, you can try to reason out what type of limits on $n_1$ and $n_2$ before the test becomes "interesting".
For example, you would probably want to avoid $n_1 = N$. For that what if you chose $N \gg n_1$ ? Is that sufficient? can you find a counter example? What if ${n_1} \approx N/2$ or $N/4$ is that okay? can you find a counter example?
For further reading have a look at Miller-Rabin primality test and Lucas-Lehmer primality test. If you are really into it you can also look at this paper $\text{Common factors of resultants}\mod p$, it may give you more interesting insights.