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**1. For any positive integer n, if 7n+4 is even, then n is even.**

2.Sum of any two positive irrational numbers is irrational.

3. If m, d, and k are nonnegative integers with d=/=0 then (m+dk) mod d = m mod

4. For all real x, if x^2=x and x=/=1 then x=0

5. If n is an integer not divisible by 3, then n^2 mod 3=1

2.Sum of any two positive irrational numbers is irrational.

3. If m, d, and k are nonnegative integers with d=/=0 then (m+dk) mod d = m mod

4. For all real x, if x^2=x and x=/=1 then x=0

5. If n is an integer not divisible by 3, then n^2 mod 3=1

**2. Basically I have to prove (or disprove) all of those and I'm stuck. Any advice and feed back would be appreciated.**

**3. Here are my attempts at solutions**

1. By contraposition, if n is odd then 7n+4 is odd. If N is odd, then n=2k+1 for some integer k. so 7(2k+1)+4, and by algebra we get 2(7k+5)+1. 7k+5 is an integer, so it must be odd.

Is this right? I think it is but I'm never sure because I'm terrible at number theory.

2. For this one I found a counterexample and I think that is sufficient to get the problem right but I want to know why it's a false statement..anyone have any insight?

3. This is confusing. Basically after half an hour of writing stuff I haven't reached any conclusions. I'm sure it has something to do with the quotient remainder theorem because the form of (m+dk) is very similar to QRT where given any interger n and positive integer d, there exists unique integers q and r such that n=dq+r. Any help would be appreciated.

4. This one is really bothering me. I know that it's true but I'm trouble proving it. I keep going back to the method of exhaustion but obviously that won't work. I have a feeling that this is really simple and I'm just overlooking something.

5. This is another that I know is true but I don't know how to prove it. I think part of the problem I'm having here is defining n as a integer not divisible by 3.

Any help on any of those would be appreciated.

Thanks

1. By contraposition, if n is odd then 7n+4 is odd. If N is odd, then n=2k+1 for some integer k. so 7(2k+1)+4, and by algebra we get 2(7k+5)+1. 7k+5 is an integer, so it must be odd.

Is this right? I think it is but I'm never sure because I'm terrible at number theory.

2. For this one I found a counterexample and I think that is sufficient to get the problem right but I want to know why it's a false statement..anyone have any insight?

3. This is confusing. Basically after half an hour of writing stuff I haven't reached any conclusions. I'm sure it has something to do with the quotient remainder theorem because the form of (m+dk) is very similar to QRT where given any interger n and positive integer d, there exists unique integers q and r such that n=dq+r. Any help would be appreciated.

4. This one is really bothering me. I know that it's true but I'm trouble proving it. I keep going back to the method of exhaustion but obviously that won't work. I have a feeling that this is really simple and I'm just overlooking something.

5. This is another that I know is true but I don't know how to prove it. I think part of the problem I'm having here is defining n as a integer not divisible by 3.

Any help on any of those would be appreciated.

Thanks

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