If n = 4k+3, does 8 divide n^2-1? Can u check if i did this right?

[mr_coffee]

ello ello!

I'm pretty sure this is right but not 100%. Directions are as follows:
GIve a reason for your answer in each of 1-13. Assume that all variables represent integers.

If n = 4k + 3, does 8 divide n^2-1? Here is my answer:
No. n^2-1 = (4k+3)^2-1 = (4k+3)(4k+3)-1 = (16k^2+24k+9)-1 = 16k^2+24k+8 = 16(k+1/2)(k+1). K+1 is an integer but k+1/2 is not necessarily an integer, because 1/2 is not an integer.

Is my reasoning correct? thanks!

 

[shmoe]

That doesn't work. Failing to write n^2-1 as 8*(some integer) doesn't mean it can't be done.

You got down to:

n^2-1=16k^2+24k+8

just fine. For what values of k is the thing on the right divisible by 8?

 

[mr_coffee]

well if i divide (16k^2+24k+8)/8 = (3k^2+12k+4)/4.
Is what what you ment? It does divide evenly, and those are all integers still. So can i say, 3k^2+12k+4 is an integer and 4 is an integer because the sums and products of integers are integers?

 

[shmoe]

well if i divide (16k^2+24k+8)/8 = (3k^2+12k+4)/4.

Something funny happened here, check this carefully.

 

[HallsofIvy]

Why did you factor out 16? You only need to show that it is 8(an integer).

 

[mr_coffee]

Sorry i had a typo it should have been, (3k^2+12k+4)/2. But I'm lost on what they want.

If n = 4k+3, does 8 divide n^2-1, well yes i just did it above. Nut I'm not sure how that showed 8 is an integer.

THe book did an example like this, but instead of n = 4k+3 its n = 4k+1. They did the following:

n^2-1 = (4k+1)^2 - 1 = (16k^2+8k+1)-1 = 16k^2+8k = 8(2k^2+k), and 2k^2+k is an integer becuase k is an integer an sums and products of integers are integers.

 

[shmoe]

Sorry i had a typo it should have been, (3k^2+12k+4)/2.

$$\frac{16k^2+24k+8}{8}=\frac{16k^2}{8}+\frac{24k}{8}+\frac{8}{8}\neq\frac{3k^2+12k+4}{2}$$

But I'm lost on what they want.

To show n^2-1 is divisible by 8, you can show (n^2-1)/8 is an integer. Equivalently, you can show there exists an integer x where n^2-1=8*x, like they did in your example (no real difference).

 

[mr_coffee]

My bad! thanks shmoe that makes sense, now I can just show:
$$\frac{16k^2+24k+8}{8}=\frac{16k^2}{8}+\frac{24k}{8 }+\frac{8}{8}$$ = 2k^2+3k+1 = (k+1)(2k+1). (k+1) and (2k+1) are both integers becuase k is an integer and the sums and products of integers are integers.

Thanks alot! :D

 

[HallsofIvy]

I was tempted to post that not being able to divide by 8 is a handicap in number theory! (but I won't!)

You don't need to factor that: Obviously, 2k²+3k+1 is an integer whether it can be factored or not.

I think it is somewhat simpler just to write 16k²+ 24k+ 8= 8(2k²+3k+1) rather than dividing by 8. The fact that it is a multiple of 8 is sufficient to prove divisibility.

 

[mr_coffee]

hah I think i am handicapped, I don't enjoy Descrete Math. This course is what is keeping me in Computer ENgineering and not Computer Science.

Thanks for the help!