If $a\oplus b=\text{L.C.M. of }a\text{ and }b$, and $a\otimes b=\text{G.C.F. of }a\text{ and }b$, then $(12\otimes 16)\oplus (24\otimes 36)$ will be:

Let's resolve the operations inside the brackets first using the custom definitions:
1. For the first term, evaluate the Greatest Common Factor (G.C.F.) of 12 and 16:

$$(12\otimes 16)=\text{G.C.F.}(12,16)=4$$

2. For the second term, evaluate the Greatest Common Factor (G.C.F.) of 24 and 36:

$$(24\otimes 36)=\text{G.C.F.}(24,36)=12$$

Now substitute these results back into the final operation:

$$(12\otimes 16)\oplus (24\otimes 36)\to 4\oplus 12$$

Applying the operation rule for $\oplus$, which calculates the Least Common Multiple (L.C.M.):

$$4\oplus 12=\text{L.C.M.}(4,12)=12$$