It's due to these properties of modulo arithmetic listed here: https://en.wikipedia.org/wiki/Modular_arithmetic#Properties

For all the operations we care about in this problem, op(a) == op(b) (mod n) if a == b (mod n). So as long as we choose an n large enough so that the divisibility check for each monkey can still be performed, we can keep taking the worry levels (mod n) to prevent them from growing too large.

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Hi. suppose that we want to check if a number is divisible by 4. 5 is not divisible as 1. in other words. if 5 is not divisible then also 1 isn't divisible. so we can check if 1 is divisible insteed of 5, and 5 is just 5-4. we can rest any divisible number and the resultant is as equal divisible as the original. what number can we substract?. the multiplication of all the divisible check numbers is always divisible by all checks. this also is the same as get de modulus