5. Now define an operation * :

When $m$ and $n$ are both positive odd numbers or both positive even numbers,
$$
m * n=m+n \text {; }
$$

When one of $m$ and $n$ is a positive odd number and the other is a positive even number,
$$
m * n=m \cdot n \text {. }
$$

Then, the number of elements in the set $M=\{(a, b) \mid a * b=36, m$ 、 $\left.n \in \mathbf{N}_{+}\right\}$ is $\qquad$ .
5.41.

Since $36=1+35=3+33=5+31=7+29=\cdots=$ $33+3=35+1$ (a total of 18) $; 36=2+34=4+32=\cdots=32$ $+4=34+2$ (a total of 17) ; $36=1 \times 36=3 \times 12=4 \times 9=9$ $\times 4=12 \times 3=36 \times 1$ (a total of 6), therefore, the number of elements in set $M$ is $18+17+6=41$.