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

Home > Numbers
10001, 100010001, .... (Posted on 2003-05-23) Difficulty: 3 of 5
Consider the following sequence:

10001, 100010001, 1000100010001, 10001000100010001, 100010001000100010001,....

Show that none of them are prime numbers.

See The Solution Submitted by Fernando    
Rating: 3.3333 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts Part of the Solution - Hint | Comment 6 of 12 |
Firstly, we see that the number 10001 is not a Prime since it can be expressed as the product of the two numbers 73 and 137. That is,
10001 = 73 * 137
Similarly, we have:
1000100010001 = 73 * 13700000137
........ and so on.

Therefore, we can see that the numbers which contain '2n' number of 1's (Ones), where 'n' is a Positive Integer is always divisible by 73 and therefore cannot be a Prime.

Secondly, we know that the numbers the sum of whose digits is divisible by 3 (or is a multiple of 3) is divisible by 3. So all the numbers of the given sequence(of the given form) containing '3n' number of 1's (Ones), where 'n' is a Positive Integer is obviously divisible by 3 and therefore not a prime.

What we have obtained till now is that the numbers of the given form which contain '2n' and '3n' number of 1's (Ones), that is 2,3,4,6,8,9,10,12,14,15,16,18,20,........ number of Ones) can never be Primes.

Now someone Please help me with the rest of the solution, that is to prove that the number with 5,7,11,13,17,19,........ number of Ones cannot be a Prime.
  Posted by Ravi Raja on 2003-05-24 05:21:04
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 (4)
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