Possible Odd Result

To get an odd number out of four even numbers, we need to use division

Note:

$42 = 2\times3\times7$

$44 = 2^2 \times 11$

$46 = 2 \times 23$

$48 = 2^4 \times 3$

We cannot put a $\div$ in front of $42, 44$ and $46$ as it will have a unique odd prime left in the denominator such that we cannot get a integer.

Therefore, the only possible solution $= 42 \times 44 \times 46 \div 48 = 7 \times 11 \times 23 = \boxed{1771}$