150 followers problem!

There is a number A B C D \overline{ABCD} such that A B C D 2 \overline{ABCD}^{2} = P Q R S A B C D \overline{PQRSABCD} .Then can you tell the only possible value of the four digit number A B C D \overline{ABCD} .

Extra credit :prove that this is the only possible answer for a 4 digit number

The answer is 9376.

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1 solution

Akshay Yadav
Sep 26, 2015

Quite an easy one,

We are searching for a 4-digit number whose square also has the same last 4 digits,

So we must begin our search from the units digit,

The numbers which have the same units digit after squaring are 0 , 1 , 5 , 6 0,1,5,6 ,

Now we extend it ten's digit 00 , 01 , 25 , 76 00,01,25,76 ,

We observe that possibility of 0 0 and 1 1 to be units digits are nill, hence we continue with 25 , 76 25,76 ,

Taking one more step ahead we end up with 625 , 376 625,376 ,

And at the final step we observe 0625 0625 and 9376 9376 .

Hence only 9376 \boxed{9376} is the answer.

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