150 followers problem!

Quite an easy one,

We are searching for a 4-digit number whose square also has the same last 4 digits,

So we must begin our search from the units digit,

The numbers which have the same units digit after squaring are $0,1,5,6$ ,

Now we extend it ten's digit $00,01,25,76$ ,

We observe that possibility of $0$ and $1$ to be units digits are nill, hence we continue with $25,76$ ,

Taking one more step ahead we end up with $625,376$ ,

And at the final step we observe $0625$ and $9376$ .

Hence only $\boxed{9376}$ is the answer.