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 Duplicate Digit Determination (Posted on 2006-12-03)
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! No Rating 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".

 Subject Author Date Answer K Sengupta 2008-11-08 00:53:38 Another solution without logs Charlie 2006-12-04 09:11:08 An easy way out Federico Kereki 2006-12-03 17:51:41 Solution without log JLo 2006-12-03 12:13:40 Some calculations (spoiler) Steve Herman 2006-12-03 11:08:27 solution Charlie 2006-12-03 11:05:20

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