Weird Divisibility

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The professor is back in class to test students. This time it is a lecture on Algebra. The professor wrote down $N$ integers on the board. Each of the $N$ integers has $M$-digits (without leading zeros). The professor also has an integer $K$ which he wrote on the board. Now he asks the students to change some digits in some of the $N$ integers. The changes are to be made such that

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solution.cppC++17

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