True False (mod n) question

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aweeks123

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For every a, b, c ∈ N, if ac ≡ bc (mod n) then a ≡ b (mod n).

I do not understand this question at all...
 
aweeks123 said:
For every a, b, c ∈ N, if ac ≡ bc (mod n) then a ≡ b (mod n).

I do not understand this question at all...
Click to expand...
What level of Mathematics class are you taking?

What is the topic of discussion - that generated this problem (HW)?
 
x ≡ y (mod n) means by definition that (x-y) is a multiply of n.
ac ≡ bc (mod n) means that (ac-bc) is a multiple of n, that is ac-bc = kn for some integer k. Does this guarantee us that a-b is a multiple of n?
Now it time for you to THINK.
 
aweeks123 said:
For every a, b, c ∈ N, if ac ≡ bc (mod n) then a ≡ b (mod n).

I do not understand this question at all...
Click to expand...
That's because you aren't stating the whole problem.

Notice that [imath]4 \cdot 5 \equiv 4 \cdot 1 \text{ (mod 8)}[/imath]. Is [imath]5 \equiv 1 \text{ (mod 8)}[/imath]? Look through whatever section you are studying and you will see that there is an extra condition on this statement.

-Dan
 
It is a true or false question. Sorry. I am coming up with true, because if c=n then ac=bc is true as well?
 
aweeks123 said:
It is a true or false question. Sorry. I am coming up with true, because if c=n then ac=bc is true as well?
Click to expand...
I don't care if it is a true or false question. You clearly don't understand what the question is asking. Do you understand when you can cancel that number c? It should be in your course materials.

-Dan
 
aweeks123 said:
It is a true or false question. Sorry. I am coming up with true, because if c=n then ac=bc is true as well?
Click to expand...
The problem was, as I understand it,

Is the following true or false?​
​
For every a, b, c ∈ N, if ac ≡ bc (mod n) then a ≡ b (mod n).​

You appear to be unfamiliar with what this sort of question means (which is not unusual for students facing their first course involving proof). They are asking whether this is always true: that whenever you have three numbers such that ac ≡ bc (mod n) then it must be true that a ≡ b (mod n).

That is why, in post #5, a counterexample was suggested - that is, an example in which ac ≡ bc (mod n), but it is not true that a ≡ b (mod n).

It was also suggested that you look at the theorems you have been taught, and think about whether any of them state (or imply) the fact you are asked about, or whether some additional condition is required (as it is). Have you done that?

When you say,
aweeks123 said:
if c=n then ac=bc is true as well?
Click to expand...
you appear to be thinking that if you take c to be n itself, then ac ≡ bc (mod n), since it is always true that an ≡ bn (mod n). Correct. But don't you see that this is reason to say that the statement is false? You've seen that the condition of the statement can be true without the conclusion having to be true.
 
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