Question

Show that if n is an integer and n^3 + 5 is odd then n is even

Answer 1

if i use contraposition, then that means i should assume n is odd first

Answer 2

and n^3 + 5 is even

Answer 3

so if n is odd...

n = 2k + 1

Answer 4

then if i substitute


(2k+1)^3 + 5

Answer 5

i assume this to be even so

(2k + 1)^3 + 5 = 2n

Answer 6

If n^3+5 is odd then n^3 is even thus n is even.

Answer 7

that doesn'r make sense....

Answer 8

Why?

Answer 9

1. how can you concur n^3 is even from n^3 + 5
2. im using contraposition

Answer 10

Sum of two odd numbers is always even......So 5 is odd and n^3+5 is also odd.Thus, n^3 must be even.

Answer 11

but n^3 + 5 is the sum....not n^3

Answer 12

Ya...... n^3+5 is the sum which is odd AND 5 is also odd....... which means n^3 must be even.

Answer 13

you said sum of two odds is even

Answer 14

Assume n odd -> n^3 odd

Answer 15

n^3 is even
5 is odd

how does that make sum of two odds

Answer 16

Actually "Assume n odd -> n^3 odd" QED.

Answer 17

Because sum of odd and even is always odd.

Answer 18

better. however, like i said im using contraposition

Answer 19

but i'll have to prove n^3 is odd when n is odd first...

Answer 20

Why use contraposition?

Answer 21

it was the specifics of the problem

Answer 22

and also to teach myself versatility

Answer 23

What kind of method is that......never heard of it.

Answer 24

assume p and q are false therefore \(\neg q \rightarrow \neg p \equiv T\)

Answer 25

if a, then b

=>

if not b, then not a

Answer 26

so i suppose my next step is to prove n^3 is odd

Answer 27

Wants to show
If n not even, then blah is odd

Going all round the houses.....

Answer 28

n^3+5=odd and we know if we add two odd num. so we get even so n^3 always be even so n=even

Answer 29

wait no...that's not the step

Answer 30

@vipul92 contraposition

Answer 31

I think that was what I did.

Answer 32

that would do : if n is odd, then n^3 is odd

Answer 33

@vipul92

Answer 34

1. how to prove n^3 is odd
2. the aim is to prove n^3 + 5 is even

Answer 35

http://answers.yahoo.com/question/index?qid=20090304175857AAXIDyP

Answer 36

so i had

(2k + 1)^3 + 5 = 2n

8k^3 + 12k^2 + 6k + 1 + 5 = 2n

8k^3 + 12k^2 + 6k + 6 = 2n

2(4k^3 + 6k^2 + 3k + 3) = 2n

let 4k^3 + 6k^2 + 3k + 3 = x

2x = 2n

QED

Answer 37

@igbasallote go through this link

Answer 38

as we know the cube of any odd is odd and for any even is even

Answer 39

@mayankdevnani i'd rather not have spoilers

Answer 40

what

Answer 41

i don't want to see the future steps. it spoils the fun

Answer 42

I think she wants to work it though....

Answer 43

she?

Answer 44

He, whatever...

Answer 45

ok

Answer 46

just because i hate math doesn't make me a girl...just saying....

Answer 47

Proof?

Answer 48

anyway...did i do the contraposition right?

Answer 49

Yes, u did (still think that's the long way round, though)

Answer 50

what about contradiction? i just assume one proposition is false right?

Answer 51

It's immediate by assuming n odd, then n^3 is odd...

Answer 52

im not as good as you in math

Answer 53

For the reason along the lines you used in the other proof..

Answer 54

anyway...proving by contradiction would be...

assume n = odd then i prove n^3 + 5 is odd

right?

Answer 55

Well u know odd by odd is odd, right (usually this can be assumed) so

If n is odd so is n^3

Answer 56

true...but as you may have noticed....i prefer going step by step...don't worry, you'll see me skipping a lot of steps when i'm sure i know proving already

Answer 57

And your question already gives n^3 +5 as odd

Answer 58

anyway...is my assumption on proving by contradiction right?

Answer 59

No, because of what I just said,,,

Answer 60

n^3 + 5 is odd?/

Answer 61

The assumption here is that n is odd and we want to see if this assumption leads to contradiction.

Answer 62

yes...

Answer 63

So the first conclusion from this assumption is that n^3 is odd.

Answer 64

yes

Answer 65

And so n^3 +5 is even (odd + odd is even), which is a contradiction.

Answer 66

yes

Answer 67

although i just asked if my assumption for contradiction was right...

Answer 68

So n is not odd, it is even QED.

Answer 69

although i just asked if my assumption for contradiction was right...

and I explained to you that the assumption is simply that n is odd

Answer 70

Problem- show n even
Assume - n odd
Find- a contradiction
Therefore - assumption is wrong

Answer 71

so my assumption was right.

Answer 72

by "assumption was right" i meant that it'sthe assumption i use for contradiction

Answer 73

U said "assume n = odd then i prove n^3 + 5 is odd"

This is incorrect.

Answer 74

well i am going to prove by contradiction..

Answer 75

so in a way my goal is to prove that n^3 + 5 is odd and contradict it

Answer 76

You cannot prove n^3 +5 is odd because n^3 +5 is given as odd already.

Answer 77

you dodn't have to take every word literally

Answer 78

I'm not sure what you want me to say, the assumption (ie something not already given) u make is that n is odd and go from there....

Answer 79

although i admit my wording is poor. i was just asking about the sense...

Answer 80

yes that's what i was asking

Answer 81

i was just asking whether i just assume one to be wrong

Answer 82

You shouldn't make any prior judgement about where your assumption will lead....

Answer 83

i didn't make any prior judgements..

Answer 84

You simply hope that it will lead to something useful for your argument.

Answer 85

If your are looking for the usual wording then it would be something like:

"For the purpose of deriving a contradiction, we assume that n is odd...."

Answer 86

that seems too formal and long...and like you've said before, you hate long methods when it can be done simply

Answer 87

True, it depends on the situation (is it a test, what can u assume, what do you have to prove, etc)

Answer 88

At some point, you leave behind the basic number properties and just concentrate on the logic and reasoning.

Answer 89

Now your current question is iff so both directions...

Answer 90

i believe in displacement over distance