So I need to prove that is divisible by 8 iff n is odd

I wrote n= 2P+1 where P = 2k

then subing 2k back in I get

16k^2+8k = 8(2k^2+k)

Is this okay?

Printable View

- March 22nd 2009, 02:15 PMmeg0529[SOLVED] Please check proof
So I need to prove that is divisible by 8 iff n is odd

I wrote n= 2P+1 where P = 2k

then subing 2k back in I get

16k^2+8k = 8(2k^2+k)

Is this okay? - March 22nd 2009, 03:27 PMReckoner
- March 22nd 2009, 03:40 PMSoroban
Hello, meg0529!

Quote:

I need to prove that is divisible by 8 iff is odd

I wrote , where . . . . no

does**not**have to be even.

can be__any__integer, and will be odd.

So we have: .

We see that is divisible by 4.

Since is the product of two consecutive integers,

. . one of them is even, the other is odd.

Hence: . is divisible by 2.

Therefore, is divisible by 8.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It said, so we must prove it "the other way."*if and only if*

. . If is divisible by 8, then is odd.

We have: . is divisible by 8.

. . Then: . , for some integer

And we have: .

We see that is even . . . hence: is odd.

. . So we have: .

It can be shown that:

. . If the square of an integer is odd, the integer is odd.

The proof is quite simple, but omitted here.

Therefore: . is odd.

- March 22nd 2009, 03:45 PMmeg0529
Reckoner,

Ok I see your point, So you are saying I need to do this is two different steps? Is there a way for me to do this in one shot?

Thanks - March 22nd 2009, 03:48 PMReckoner
Certainly there is, and Soroban has provided it.

Take care.

Edit: Oh, and be sure you understand what is meant by "if and only if" (abbreviated iff). I misread your post initially and thought it only went one direction; there are basically two proofs that need to be made. Again, Soroban has this covered. - March 22nd 2009, 04:44 PMmeg0529
Got it. Thanks you both very much

- March 22nd 2009, 05:22 PMReckoner
I see you have edited your post, but I will post my response anyway since it is already written.

No. An if-then statement means the implication goes in only one direction.

The statement "If then " which can also be written as only if means that if is true, then must also be true (but it might be possible for to be true even though is false).

The statement if means that if is true, then must also be true (but it might be possible for to be true even though is false).

The statement if and only if means that if is true, then must also be true,*and*if is true, then must also be true. Do you see how the implication goes both ways?

When proving a statement like you need to prove both of these statements:

1. If then

2. If then - March 22nd 2009, 07:16 PMmeg0529
Ya haha I re-read the chapter, thanks!