Michael's number

Algebra Level 3

For a positive four-digit integer \(n,\) let \(T(n)\) be the number created by swapping the hundreds and thousands digits of \(n\) and swapping the tens and units digits of \(n.\) There is a unique integer \(M\) such that \(T(M)= 4M.\) What are the last three digits of \(M\)?

This problem is posed by Michael T.

Details and assumptions

For example, \(T(1234)=2143\) and \(T(1508) = 5180.\)

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