Is the language L = {(a,b)* | #a * #b is an odd number} regular?
It is.

Since only the product of two odd numbers is an odd number, you can construct a DFA that only accepts when #$a$ and #$b$ are (both) odd.

The followings states would be:


$Q_1$ : #$a$ even, #$b$ even
$Q_2$ : #$a$ even, #$b$ odd
$Q_3$ : #$a$ odd, #$b$ even
$Q_4$ : #$a$ odd, #$b$ odd (an accepting state)


The transitions are fairly simply, and I'll leave them to you.