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        riddles >> easy >> 444
        
(Message started by: FiBsTeR on Mar 13th, 2009, 7:30pm)
Title: 444
Post by FiBsTeR on Mar 13th, 2009, 7:30pm
The square of 38 is 1444, which ends with three fours. What is the smallest positive integer greater than 38 with this property?

No calculators/computers!
Title: Re: 444
Post by towr on Mar 14th, 2009, 4:30am
[hide](38+n)2 = 1444 + 76n + n2
76n + n2 = 0 (mod 1000)
n (76+n) = 0 (mod 1000)
n=424
38+424 = 462[/hide]
Title: Re: 444
Post by rmsgrey on Mar 14th, 2009, 5:01am
Different approach from towr, but same answer ([hide]462[/hide]):

[hideb]
(x+38 )(x-38 ) = x2-382
so
1000|(x+38 )(x-38 )
4|(x+38 ) or 4|(x-38 )
either way, x=2 (mod 4)

5 can't divide both x+38 and x-38, so
125|(x+38 ) or 125|(x-38 )
x=87 (mod 125) or x=38 (mod 125)

Combining the two constraints gives two numbers mod 500:

38 and 462

A quick calculation to check shows 4622=213444[/hideb]


...edited to remove 8) smileys...


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