# 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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