# Divisibility with Base 16

Number Theory Level pending

Let $$n=11335577_{16}$$. Let $$S_T$$ be defined as the set of all distinct base-16 numbers that are permutations of $$n$$ (for example, $$13135757$$, $$13571357$$, $$11335577$$ and $$77553311$$ are all part of $$S_T$$). If an element in $$S_T$$ (let's call it $$m$$) that satisfies the condition $$m-n \equiv 0 \pmod{10}$$ in base ten is also in the set $$S_M$$, find the last four digits of the sum of all values in $$S_M$$ in base ten.

For example, $$13135577_{16}$$ would be in $$S_M$$ because when both $$13135577_{16}$$ and $$n$$ are converted into base 10 ($$1 \cdot 16^7 + 3 \cdot 16^6 + \ldots + 7 \cdot 16^1 + 7 \cdot 16^0$$ and $$1 \cdot 16^7 + 1 \cdot 16^6 + \ldots + 7 \cdot 16^1 + 7 \cdot 16^0$$, respectively), the difference between these corresponding base-10 values is divisible by 10.

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