What about I randomly generate a number, add it to the illegal number to get a result. Is the resulting number illegal or not? (Thinking about it mathematically

What about I randomly generate a number, add it to the illegal number to get a result. Is the resulting number illegal or not? (Thinking about it mathematically, the resulting number's distribution isn't really affected by the illegal number - Maybe we can practically find something by taking a lot of generated numbers and analyzing the apparent distribution) Another idea is to generate a random number and multiply it with an illegal prime. If the illegal prime is a sufficiently large number; we can extract the original illegal prime with very high confidence by just finding its prime factors and picking the shortest one.