Explanation:

Let Length = l , breadth = b , and height = h
Area of 1st adjacent face = Length × Breadth
                             15 cm² = l × b                           -------------(1)
So,In this case length can be 1 or 15 or 5 or 3
     breadth can be 1cm or 15cm or 5cm or 3cm
Area of 2nd adjacent face = Breadth × Height
                              20 cm² = b × h                         --------------(2)
In this case breadth could be 1cm or 20cm or 5cm or 4cm or 10cm or 2cm
In both the cases 5 is common for breadth.
  So, breadth is 5 cm
                     Put breadth= 5 cm in (1) we get,
                                 15cm² = l × 5cm
                               ⇒ length = 15cm²/5cm = 3cm 
and 
                         Putting breadth = 5cm in (2) we get,
                         20cm² = 5cm × height
                         height= 20cm²/5cm = 4cm 
So, Length   = 3 cm
       Breadth = 5 cm
       Height   = 4 cm
We know that volume of cuboid is Length×Breadth×Height
So, Volume of cuboid = 3cm×5cm×4cm =60cm³ 

Hence, option 4 is the correct answer.