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Duplicate Digit Determination (Posted on 2006-12-03) Difficulty: 3 of 5
If 2^P and 5^P start with the same (non-zero) digit for positive integer P, what is that digit? Can you prove it must be so?

  Submitted by Old Original Oskar!    
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Solution: (Hide)
A not very hard way: Since 2^P times 5^P is 10^P, the digit must be 1 or 3; it is easy to see that 1 won't work (if both 2^P and 5^P start with 1, the product cannot be 10^P) so the answer is 3. Of course, this doesn't prove that 2^P and 5^P ever start with the same digit, but see the second solution below...

An even easier way: 2^5 and 5^5 both start with 3, so if there is a single answer, it must be "3".

Comments: ( You must be logged in to post comments.)
  Subject Author Date
AnswerK Sengupta2008-11-08 00:53:38
SolutionAnother solution without logsCharlie2006-12-04 09:11:08
SolutionAn easy way outFederico Kereki2006-12-03 17:51:41
SolutionSolution without logJLo2006-12-03 12:13:40
SolutionSome calculations (spoiler)Steve Herman2006-12-03 11:08:27
SolutionsolutionCharlie2006-12-03 11:05:20
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