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

Home > Just Math
N-Divisibility II (Posted on 2011-12-17) Difficulty: 3 of 5
Consider four base ten positive integers 4298, 7070, 10542 and 15428 and, determine the total number of positive integers dividing:

(I) At least one of the four given numbers.

(II) At least two of the four given numbers.

(III) At least three of the four given numbers.

(IV) Each of the four given numbers.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
2nd things 3rd (partial spoiler) | Comment 3 of 5 |
The four numbers are (2*3*7)^98, (2*5*7)^70, (3*5*7)^42, and (2*7*11)^28.

Part (II) -- 
a) Any number which divides the 1st and 2nd must divide (2*7)^70. 

b) Any number which divides the 1st and 3rd must divide (3*7)^42. 

c) Any number which divides the 1st and 4th must divide (2*7)^28. 

d) Any number which divides the 2nd and 3rd must divide (5*7)^42. 

e) Any number which divides the 2nd and 4th must divide (2*7)^28. 

f) Any number which divides the 3rd and 4th must divide 7^28.

We can ignore cases c, e and f, because they are all subsets of a.

So, we are looking for all numbers which divide (2*7)^70 or (3*7)^42 or (5*7)^42.

To compute these, 
add
1) The number which divide 7^70 (ie, 71)
2) The number which divide (2*7)^70 but which are not a pure power of 7 = (70 powers of 2) * (71 powers of 7) = 4970
3) The number which divide (3*7)^42  but which are not a pure power of 7 = (42 powers of 3) * (43 powers of 7) = 1806
4) The number which divide (5*7)^42  but which are not a pure power of 7 = (42 powers of 5) * (43 powers of 7) = 1806 

Total number = 71 + 4970 + 1806 + 1806 = 8653

No time now to solve the 1st part.  Perhaps somebody else will check my work, and finish the problem

  Posted by Steve Herman on 2011-12-17 12:47:51
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 (7)
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