You are given a string num representing a positive integer, and an integer t.

Return a string representing the smallest integer that is greater than or equal to num, contains no $0$ digits, and whose digit product is divisible by t. If no such integer exists, return "$-1$".

Constraints:

• $2 \leq$ num.length $\leq 2 imes 10^5$

• num consists only of digits in the range ['$0$', '$9$'].

• num does not contain leading zeros.

• $1 \leq$ t $\leq 10^{14}$

Now, let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

We have a game for you to play. Rearrange the logical building blocks to develop a clearer understanding of how to solve this problem.

Implement your solution in the following coding playground.