Can someone help me prove or disprove this statement:

For all integers a,b, and c, if a divides b (and results in some integer) and a cannot divide c (and result in some integer) then a cannot divide b+c (and result in some integer).

I have done a lot of work on this and im having a hard time, I did not find out much except that :

if a|b and b|c then a|c by transitivity
also, if a|b and b|c then a|b+c.

Thank you

2. Originally Posted by matthayzon89
Can someone help me prove or disprove this statement:

For all integers a,b, and c, if a divides b (and results in some integer) and a cannot divide c (and result in some integer) then a cannot divide b+c (and result in some integer).

I have done a lot of work on this and im having a hard time, I did not find out much except that :

if a|b and b|c then a|c by transitivity
also, if a|b and b|c then a|b+c.

Thank you
Dear matthayzon89,

Suppose that,

$a\mid{(b+c)}$

Also it is given that, $a\mid{b}$

Therefore, $a\mid{c}$

Therefore, a does not divide b+c.

3. Originally Posted by Sudharaka
Dear matthayzon89,

Suppose that,

$a\mid{(b+c)}$

Also it is given that, $a\mid{b}$

Therefore, $a\mid{c}$

Therefore, a does not divide b+c.

Hello,
Thanks for the response.

How did you conclude that it is a contradiction? Unfortunately, I don't really understand your explanation...

My logic, (which is probably incorrect) is that the problem tells us that you can divide b by a and get an integer it also says that you cannot divide c by a to get an integer. Then how can that conclude that you cannot divide b+c by a and get an integer? wouldn't this statement be false?

Can I prove that this false through counter example? For example if I found a scenario where I can divide b+c by a and get an integer, and I can divide b by a but I cannot divide a by c...

Can you explain how you would prove something using contradiction?

4. Problem: For all integers a, b, and c, if a divides b (and results in some integer) and a cannot divide c (and result in some integer) then a cannot divide b+c (and result in some integer).

This is what I came up with, Can someone let me know if it is correct?

Suppose that,

b+c/a

this can also be written as b/a + c/a.

We know that b/a IS an integer, we also know that c/a IS NOT an integer.

Therefore, b/c+c/a cannot be an integer. (Integer+non-integer= non integer).

Therefore this statement is true. Is it not?

5. Originally Posted by matthayzon89
Hello,
Thanks for the response.

How did you conclude that it is a contradiction? Unfortunately, I don't really understand your explanation...

My logic, (which is probably incorrect) is that the problem tells us that you can divide b by a and get an integer it also says that you cannot divide c by a to get an integer. Then how can that conclude that you cannot divide b+c by a and get an integer? wouldn't this statement be false?

Can I prove that this false through counter example? For example if I found a scenario where I can divide b+c by a and get an integer, and I can divide b by a but I cannot divide a by c...

Can you explain how you would prove something using contradiction?
Dear matthayzon89,

I assumed that a divides b+c then using that assumption and the given facts in the problem I have showed that a divides c which is not true. Therefore it could be concluded that my assumption is incorrect. This is know as the contradiction method. First you make and assumption and using that assumption and the things that are given in the problem if you could come to a solution which is incorrect, (either a obvious fact such as 2=4 or else a given fact in the problem, as in this case) then you could tell that your assumption is wrong.

You cannot use the counter example method, because you have to prove or disprove the statement for a general situation. If you give a counter example that means the statement is not true for that instance. But it does not tell that a does not divide b+c for all values that you could take for a,b and c.

If you have any doubts you may always ask them.

6. Originally Posted by matthayzon89
Problem: For all integers a, b, and c, if a divides b (and results in some integer) and a cannot divide c (and result in some integer) then a cannot divide b+c (and result in some integer).

This is what I came up with, Can someone let me know if it is correct?

Suppose that,

b+c/a

this can also be written as b/a + c/a.

We know that b/a IS an integer, we also know that c/a IS NOT an integer.

Therefore, b/c+c/a cannot be an integer. (Integer+non-integer= non integer).

Therefore this statement is true. Is it not?
Dear matthayzon89,

Your way of proof is correct as far as I am concerned. In my proof I had used the basic concepts that we learnt for Abstract algebra.