How Do You Solve 7x ≡ 3 (mod 15)?

  • Thread starter Thread starter hammonjj
  • Start date Start date
  • Tags Tags
    Equivalence
Click For Summary

Homework Help Overview

The discussion revolves around solving the congruence equation 7x ≡ 3 (mod 15). Participants are exploring the implications of this equation and the methods to find values of x that satisfy it.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • One participant attempts to manipulate the equation into a different form, expressing it as 7x - 3 = 15n, but struggles with the factoring approach. Others question the phrasing of the problem and suggest that the goal may be to find specific values of x rather than proving a general statement.

Discussion Status

Participants are actively engaging with the problem, with some providing insights into the nature of the equation and suggesting methods to express 3/7 in the modular system. There is a recognition of the need to clarify the problem's intent and the conditions under which the equation holds true.

Contextual Notes

Some participants note that the equation does not hold for all values of x, prompting a discussion about the specific cases where it may be valid. There is also mention of the need to verify results, indicating a cautious approach to the problem.

hammonjj
Messages
32
Reaction score
0

Homework Statement


Show that:

7x≈3 mod(15)


Homework Equations


From the given above I think it should be:

7x-3=15n

The Attempt at a Solution


I tried factoring this in various ways to show that either said was a factor of the other, but I'm struggling here.

But I don't know what to do from here. I actually have several of these problems, but I assume that once I know how to do the first one, they will be easy.

Thoughts? Thanks!
James
 
Physics news on Phys.org
hammonjj said:

Homework Statement


Show that:

7x≈3 mod(15)

Homework Equations


From the given above I think it should be:

7x-3=15n

The Attempt at a Solution


I tried factoring this in various ways to show that either said was a factor of the other, but I'm struggling here.

But I don't know what to do from here. I actually have several of these problems, but I assume that once I know how to do the first one, they will be easy.

Thoughts? Thanks!
James

Well, for starters, it isn't true in general (for all x).

counterexample: for x = 3, 7*3 = 21 = 6 mod 15
 
So, since the equation isn't generally true, maybe the aim of the problem was to find the values of x for which it is true.
 
Mark44 said:
So, since the equation isn't generally true, maybe the aim of the problem was to find the values of x for which it is true.

In which case "Show that:", etc. is a terrible phrasing for it.
 
Yes, it is!

Hammonjj, you want to solve 7x= 3 (mod 15) for x. Of course, that is the same as x= (3/7) (mod 15) so you really just want to know how to write 3/7 in this mod 15 system.

Notice that 7(2)= 14= -1 (mod 15) so that 7(-2)= 1 (mod 15). And, since 15- 2= 13, 1/7= -2= 13 (mod 15). Now, what is 3/7 (mod 15)?
 
hammonjj said:

Homework Statement


Show that:

7x≈3 mod(15)

Homework Equations


From the given above I think it should be:

7x-3=15n

The Attempt at a Solution


I tried factoring this in various ways to show that either said was a factor of the other, but I'm struggling here.

But I don't know what to do from here. I actually have several of these problems, but I assume that once I know how to do the first one, they will be easy.

Thoughts? Thanks!
James

Always verify with 0
 

Similar threads

  • · Replies 15 ·
Replies
15
Views
2K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 16 ·
Replies
16
Views
3K
Replies
14
Views
2K
  • · Replies 4 ·
Replies
4
Views
2K
Replies
2
Views
3K
  • · Replies 3 ·
Replies
3
Views
2K
Replies
4
Views
8K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K