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

Home > Just Math
Another Remainder Puzzle (Posted on 2007-01-25) Difficulty: 2 of 5
Given that a, b and c are positive integers satisfying the equation:

3a + 5b = 7c + 1.

Derive mathematically the possible remainders when a and c are separately divided by 4.

  Submitted by K Sengupta    
Rating: 4.0000 (1 votes)
Solution: (Hide)
Since, each of a, b and c are positive, reducing both sides of the given equation 3^a + 5^b = 7^c + 1 to Mod 3; we obtain:

2 ^b = 2 ( Mod 3 ); so that b must be odd.

Hence, 3^a + 5 = 7^c + 1 ( Mod 8).

Since, 3^a = 1, 3 (Mod 8) and 7^c = 1,7 (Mod 8); it follows that :
(3^a Mod 8, 7^c Mod 8) = (3,7), so that both a and c must be odd.

Otherwise, if at least one of a and c is even, if follows that:
(3^a Mod 8, 7^c Mod 8) = (1,7); (1,1), (3,1).
None of these three cases satisfy the relationship:
3^a +5 = 7^c + 1 (Mod 8), and accordingly, this is a contradiction.

Now, each of a, b and c are positive, and so, the equation 3^a + 5^b = 7^c + 1 gives:
3^a + 5 = 7^c + 1 (Mod 10).

Since, both a and c are odd; this is possible if and only if a = 4p-3 and
c = 4q -3 for positive integers p and q.

Consequently, dividing a and c separately by 4 we obtain the respective remainders as 1 and 1.

------------------------------------------------------------------

NOTE: When each of a, b and c are positive integers the minimum possible solution to the given equation is (a,b,c) = (1,1,1). This is also pointed out by Ady TZIDON in the Comments section.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre: A partial solution? (spoiler) ....what if...Ady TZIDON2007-01-30 09:55:05
A partial solution? (spoiler)Steve Herman2007-01-26 09:22:44
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