Friday, 23 February 2018

number theory - last digit is a square and.....



I've found some solutions for this questions but they were not impressive.



Question:



How many natural numbers are there in base $10$,whose last digit is perfect square,combination of last two digits is a perfect square,combination of last three digits is a perfect square,$\ldots$,combination of last $n$ digits is a perfect square?




For example $64$ is a number whose last digit is a perfect square and combination of last two digits is also a perfect square.



Kindly tell me how to approach this question.


Answer



Answers are in the forms:-



(i)$4\times10^n$



(ii)$9\times10^n$




(iii)$10^n$



(iv)$49\times10^n$



(v)$64\times10^n$



(vi)$81\times10^n$



Where $n \in 0,2,4,6...$


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