Divisibility problem cant use module just induction

  • Thread starter Willy_Will
  • Start date
  • #1
15
0

Main Question or Discussion Point

Hi all,

Great forum, I have been reading some cool stuff here for about a month.

Heres my question:
Using induction prove that 5 divides 8^n - 3^n, n Natural Number.

I know its true for n = 1, but I get stuck on the n = k+1. I dont know how to proceed from here: 8(8^k) - 3(3^k)

Also, I KNOW this is very easy using mod 5 but I can't do it here, I HAVE to prove it using only induction.

Any hints?

Thanks,

-William
 

Answers and Replies

  • #2
matt grime
Science Advisor
Homework Helper
9,395
3
If 5 divides X and you show that 5 divides Y-X, does 5 divide Y?
 
  • #3
15
0
I think so, but how can I use that in my problem?
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,810
934
The fact that 8= 5+ 3 is very useful here!
 
  • #5
15
0
I think I've got it.

Suppose that 5 divides 8^k - 3^k ---> 8^k - 3^k = 5x, for some integer x.

case n = k +1
= 8^k+1 - 3^k+1
= 8(8^k) - 3(3^k)
= 8(8^k) - (3^k)(8) + (3^k)(8) - (3)(3^k)
= 8(8^k - 3^k) + 3^k(8 - 3)
(1) (2)

(1) by hypotesis 5 divides the expresion.
(2) 5 = 8 - 3, 5 divides 5.

= 8(5x) + 3^k(5)
= 5(8x + 3^k)

5 divides the expresion

Please tell me its that way, if not, point the errors and some hints.

Thanks, really.

-William
 
Last edited:
  • #6
15
0
I think I've got it.

Suppose that 5 divides 8^k - 3^k ---> 8^k - 3^k = 5x, for some integer x.

case n = k +1
= 8^k+1 - 3^k+1
= 8(8^k) - 3(3^k)
= 8(8^k) - (3^k)(8) + (3^k)(8) - (3)(3^k)
= 8(8^k - 3^k) + 3^k(8 - 3)
(1) (2)

(1) by hypotesis 5 divides the expresion.
(2) 5 = 8 - 3, 5 divides 5.

= 8(5x) + 3^k(5)
= 5(8x + 3^k)

5 divides the expresion

Please tell me its that way, if not, point the errors and some hints.

Thanks, really.

-William
 
  • #7
15
0
sorry for the double post.

(1) = (8^k - 3^k)
(2) = (8 - 3)

(1) by hypotesis 5 divides the expresion.
(2) 5 = 8 - 3, 5 divides 5.

= 8(5x) + 3^k(5)
= 5(8x + 3^k)

5 divides the expresion
 
  • #8
HallsofIvy
Science Advisor
Homework Helper
41,810
934
By George, I think he's got it!
 
  • #9
15
0
Thanks guys, I really appreciate it!!
 

Related Threads on Divisibility problem cant use module just induction

  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
2
Views
2K
Replies
11
Views
938
  • Last Post
Replies
2
Views
2K
Replies
6
Views
22K
Replies
2
Views
3K
  • Last Post
Replies
14
Views
4K
  • Last Post
Replies
3
Views
3K
Replies
1
Views
3K
  • Last Post
Replies
2
Views
3K
Top