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Find the greatest number $k$ such that there exists a perfect square that is not a multiple of 10, with its last $k$ digits the same

I could find $12^2 = 144$, $38^2 = 1444$, $62^2 = 3844$ and $88^2 = 7744$

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wwdbhjcv is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
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The numbers with this property is infinitly and there is much to say: the last digits in the squares is zero or 4.. if its end in 4 then its cant be more than 3times(3 fours).there is only 4 base numbers in this properety: n=12,38,62,88 but each number of the form n+ 100k alsoend in 44 or 444

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rooh byda is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
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  • $\begingroup$ Nice claim. Prove it. $\endgroup$ Nov 25 at 16:59
  • $\begingroup$ It is an easy prove.each number n can written n=a+10b+100c, where is a,b are the last to digits and c is the other concatinate digits, then n^2=(a+10b+100c)^2 =a^2+20ab+100(b^2+2ac+20bc+100c^2) we note that the 2 end digit in n^2 is not relating to c only relate to a and b,then must test only the case when 2 end digits of interval[0,99]^2 are the same. Its happen when k= 12, 38, 62, 88 if n^2 has the same end 2 digits then must be n=k+100c $\endgroup$
    – rooh byda
    Nov 25 at 18:00
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By examination of squares considered $\bmod 1000$, the only possibility of three repeated non-zero digits terminating a square is $444$, arising from values $\{38, 462, 538, 962\} \bmod 1000$. Due to your finding for two repeated end digits, searching for these can just consider the values you found with added multiples of $100$.

Extending this to consider squares of these values with added multiples of $1000$, we find that these give squares ending in $\{1444, 3444, 5444, 7444, 9444\}$ - that is, there are no squares that end with $4$ similar non-zero digits.

This gives $k=3$.

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  • $\begingroup$ It's also easy to show that $x^2 \equiv (500-x)^2 \equiv (500+x)^2 \equiv (1000-x)^2 \bmod 1000$, meaning that only values up to $250$ need to be checked, with the other possibilities inferred from the above, in the first step above for three similar digits. A corresponding argument also applies to the search for four similar digits. $\endgroup$
    – Joffan
    Nov 26 at 6:42

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