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Topic: 444 (Read 257 times) |
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FiBsTeR
Senior Riddler
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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!
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towr
wu::riddles Moderator
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Some people are average, some are just mean.
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Re: 444
« Reply #1 on: Mar 14th, 2009, 4:30am » |
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(38+n)2 = 1444 + 76n + n2
76n + n2 = 0 (mod 1000)
n (76+n) = 0 (mod 1000)
n=424
38+424 = 462
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Wikipedia, Google, Mathworld, Integer sequence DB
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rmsgrey
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Re: 444
« Reply #2 on: Mar 14th, 2009, 5:01am » |
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Different approach from towr, but same answer (462):
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(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 |
...edited to remove  smileys...
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| « Last Edit: Mar 14th, 2009, 5:02am by rmsgrey » |
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