Puzzle ZWGZ 

Workings

Hint
It will take a while to check 1 x 7, 2 x 7, 3 x 7. What do we know about the answer?

Answer
63,492 x 7 = 444,444

Reasoning
We can work backwards, as we know that the number multiplied by 7 has only 4's. We can try the following:
      4 ÷ 7 ≈ 0.57     x
     44 ÷ 7 ≈ 6.29     x
    444 ÷ 7 ≈ 63.43    x
  4,444 ÷ 7 ≈ 634.86   x
 44,444 ÷ 7 ≈ 6,349.14 x
444,444 ÷ 7 = 63,492