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.


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

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.