Calculate the Number, Divisors Edition
Jelly, 17 16 bytes

×€ȷ5R¤ÆDL€€Z=Ḅi3


This is a brute force solution that tries all possible values up to 100,000. Try it online!

Non-competing version

The latest version of Jelly has a bug fix that allows to golf down the above code to 15 bytes.

ȷ5R×€³ÆDL€€=Ḅi3


Try it online!

How it works

×€ȷ5R¤ÆDL€€Z=Ḅi3  Main link. Left input: x,z. Right input: w,y

     ¤            Combine the two atoms to the left into a niladic chain.
  ȷ5              Yield 100,000 (1e5).
    R             Apply range. Yields [1, ..., 100,000].
x€                Multiply each r in the range by x and z.
                  This yields [[x, ..., 100,000x], [z, ..., 100,000z]].
      ÆD          Compute the divisors of each resulting integer.
        L€€       Apply length to each list of divisors.
                  This counts the divisors of each integer in the 2D array.
           Z      Zip; group the divisors of kx and kz in pairs.
            =     Compare each [divisors(kx), divisors(kz)] with [w, y].
                  This yields a pair of Booleans.
             Ḅ    Convert each Boolean pair from binary to integer.
              i3  Find the first index of 3. Yields 0 for not found.