# Thread: Prove or disprove the following statement; modulo, variables, integers, etc.

1. ## Prove or disprove the following statement; modulo, variables, integers, etc.

I am working through a past exam as part of my study for uni, and I've come across a question that I can't work out.

Prove or disprove the following statement:
For all intergers n, 3(n^2)-5 is either congruent to 2 modulo 4 or congruent to 3 modulo 4.

I really don't even know where to start with this question, I have rearranged the two modulo terms into 2mod4=4k+2 (for some integer k) and 3mod4=4p+3 (for some integer p), but have no idea whether this is of any help in solving the question or not.

Any help would be greatly appreciated,
Thanks.

2. ## Re: Prove or disprove the following statement; modulo, variables, integers, etc.

Originally Posted by lucaswilson94
I am working through a past exam as part of my study for uni, and I've come across a question that I can't work out.

Prove or disprove the following statement:
For all intergers n, 3(n^2)-5 is either congruent to 2 modulo 4 or congruent to 3 modulo 4.

I really don't even know where to start with this question, I have rearranged the two modulo terms into 2mod4=4k+2 (for some integer k) and 3mod4=4p+3 (for some integer p), but have no idea whether this is of any help in solving the question or not.

Any help would be greatly appreciated,
Thanks.
Hint: What possible values can n^2 take a modulo 4?

-Dan

3. ## Re: Prove or disprove the following statement; modulo, variables, integers, etc.

If n is even, that is, if $n= 2k$ for some integer k, then $3n^2- 5= 3(4k^2)- 5$.
If n is odd, that is, if $n= 2k+ 1$ for some integer k, then $3n^2- 5= 3(2k+ 1)^2- 5= 3(4k^2+ 4k+ 1)- 5$.

Factor "4"s out of those.

4. ## Re: Prove or disprove the following statement; modulo, variables, integers, etc.

Hi,
Here are some facts about congruence that apply to your problem. They also apply to many other problems involving congruence.