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Taking the [lateral-thinking] tag somewhat seriously, I remark that So


My answer is


Following @Vepir's answer, this is an attempt to understand if this solution is unique or if we can expect other similar solutions. So what we need to get a handle on is why 1111 divides 6^5+1; in other words, why 6^5 = -1 mod 1111. The first thing to observe is 1111 = 11x101 with 11 and 101 prime numbers. Time for Fermat: n^10 = 1 mod 11 and n^100 = 1 mod ...


All possible ways to start using one operation (two instances of $x$) are: $0,1,2x,x^2,x^x,11x$. I've built on top of $x^x$ to get a four digit number:


I have found a number can be achieved by using the same number of another digit, kind of based on OP's example Stretching this a bit, the best I've managed so far for the bonus is

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