Four digit!

If $10^4$ divides $n(n-1)$ , the fact that $n$ and $n-1$ are relatively prime implies that one of them is divisible by $2^4$ and the other is divisible by $5^4$ . This leads to two possibilities for $n$ mod $10000$ : $n \equiv 625$ and $n \equiv 9376$ . The only four-digit number that satisfies one of these congruence conditions is $\fbox{9376}$ .

Looking at the last digit, the last digit must be either 0, 1, 5 or 6.

Then looking at the last two digits, the last two digits must be either 00, 01, 25 or 76.

Then looking at the last three digits, the last three digits must be either 000, 001, 625 or 376.

Then looking at the last four digits, the last four digits must be either 0000, 0001, 0625 or 9376.

Out of those, only 9376 is a 4 digit number.