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        riddles >> medium >> Take Non-Zero Decimal Digits, Get  Square?
        
(Message started by: K Sengupta on Jul 27^th, 2007, 8:01am)

Title: Take Non-Zero Decimal Digits, Get Square?
Post by K Sengupta on Jul 27^th, 2007, 8:01am

Analytically determine whether any 9 digit decimal number that contains all the decimal digits except zero and does not end with 5 can be a perfect square.

Title: Re: Take Non-Zero Decimal Digits, Get Square
Post by SMQ on Aug 14^th, 2007, 6:00am

I don't know about analytically, but empirically, [hide]yes, there are 30 of them[/hide].

--SMQ

Title: Re: Take Non-Zero Decimal Digits, Get Square
Post by Eigenray on Aug 14^th, 2007, 9:32am

But there are none that do end in 5:

[hide]If n² ends in 5, then n=10k+5 for some k. But then n² mod 1000 is either 025, 225, or 625 depending on the vaue of k mod 5. Since the first two are unacceptable, we find that k=2 mod 5. But in this case we find that n² is either 0625 or 5625 mod 10000, which is no good.[/hide]