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This question already has an answer here:

I am learning number theory and trying to understand how does below statement is true .

Show that no integer in the following sequence can be a perfect square:
99, 999, 9999, 99999, ...
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marked as duplicate by Dietrich Burde, Parcly Taxel, Smylic, Jeremy Rickard, erfink May 15 '17 at 9:00

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  • $\begingroup$ -1 Title is misleading. $\endgroup$ – Kenny Lau May 15 '17 at 7:05
  • $\begingroup$ Change your question's title. $\endgroup$ – DonAntonio May 15 '17 at 7:09
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Notice that all perfect squares satisfy $x^2 \equiv 0,1($mod $4)$, meaning the remainder of perfect squares when divided by $4$ is always $0$ or $1$. On the other hand, all these numbers you gave have remainder $3$ when divided by $4$, so they can't be perfect squares.

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