All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
West Side Story (Posted on 2004-02-15) Difficulty: 4 of 5
Does 9 appear as the leftmost digit in the decimal representation of any power of 2?

Does 7 appear as the leftmost digit in the decimal representation of any power of 37?

Is it possible to find a power of any counting number that has a given digit as its leftmost digit?

Also, is it possible to find a power of any counting number that begins with a given series of digits (e.g., is there a power of 24 that begins with 937)?

Prove that this is possible, or give a counter-example.

Bonus: What percentage of the powers of 2 have 1 as their leftmost digit?

Note: In finding the powers of "any counting number," exclude powers of ten, whose leftmost digit is always 1.

No Solution Yet Submitted by DJ    
Rating: 3.2500 (8 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Slide Rule logic (spoiler) Comment 7 of 7 |
All of these questions are simple when thinking in terms of a slide rule.  For any number, counting or otherwise, if the log base 10 is irrational, then the set of all of its powers completely cover the slide rule, with equal density.

That makes the answers as follows.

a) Yes.  the percentage of powers of 2 that begin with a 9 = log(10) - log(9) = 1 - log(9)

b) Yes.  The percentage of powers of 37 that start with 7 = log(8) - log(7)

c) Yes, unless the counting number is a power of 10, in which case its logarithm is rational.  All powers of a power of 10 begin with a 1.

Bonus) log(2), approximately 30.1% 

Edited on July 16, 2012, 2:59 pm
  Posted by Steve Herman on 2012-03-04 11:22:42

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information