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Consider a question, that xyxyxyxy cannot be a perfect square. How should i tackle this problem. All i use is it must be $0,1 ($mod $3,4)$ and then the math, are there any another beatiful ways because this method does not work everytime.

Thank you for your time.

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Your number is $xy \cdot 1010101$. Can you find a factor that cannot be repeated?

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  • $\begingroup$ Another approach is to use the divisibility test for $11$ and then some case checking.But your method is definitely more elegant.Although I'm not sure this "works every time",but what does? $\endgroup$ – rah4927 Apr 23 '14 at 15:30
  • $\begingroup$ Thank u for the beatiful answer, but i wanted to know how to tackle(different approaches) questions like this $\endgroup$ – Shobhit Apr 23 '14 at 15:41
  • $\begingroup$ @Shobhit,can you be more explicit?I am not sure what the definition of "problems like this" is.You gain the ability to solve problems through experience.Not everything has a general method. $\endgroup$ – rah4927 Apr 23 '14 at 15:53
  • $\begingroup$ @Shobhit: when you have digit patterns, it is often useful to represent them algebraically. Sometimes it would help to replace $xy$ by $(10x+y)$ One general thing from this answer is if you have $1+x+x^2+\dots x^n$ with $n+1=pq$ you can always factor it as $(1+x+x^2+\dots x^{p-1})(1+x^p+x^{2p}+\dots x^{p(q-1)}$. If $p \neq q$ that gives two different factorizations. $\endgroup$ – Ross Millikan Apr 23 '14 at 16:30

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