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# Is the positive integer n odd?

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Is the positive integer n odd?  [#permalink]

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01 Jul 2016, 05:28
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Is the positive integer n odd?

(1) $$n^2+(n+1)^2+(n+2)^2$$ is even
(2) $$n^2-(n+1)^2-(n+2)^2$$ is even

OG Q 2017 New Question(Book Question: 194)

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Re: Is the positive integer n odd?  [#permalink]

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29 Apr 2017, 16:53
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guenthermat wrote:
stonecold wrote:
Nice Question
Here we can use the property that POWER does not effect the even or odd nature of any integer
Hence statement 1=> 3n+odd=even => n is odd=> suff
Statement 2=> -n-5=even=> n=even-odd=> odd => suff

Smash that D

stonecold - I am quite a bit lost on this question. Could you just explain a lil bit more how you get to "3n + odd = even" / "-n-5=even"?

p

Hi
Here is what i would do in this question =>

Lets start from scratch -->

Given data => n is a positive integer.
We are asked if it is odd or not.Hence we just need the even/odd nature of n.

Lets roll =>

Statement 1 =>
n^2+(n+1)^2+(n+2)^2 is odd.
There are three ways to tackle this one.
First way => Open the brackets using (a+b)^2=a^2+b^2+2ab
n^2+(n+1)^2+(n+2)^2=> n^2+(n^2+1+2n)+(n^2+4+4n)=> 3n^2+6n+5

So 3n^2+6n+5 =even
3n^2 +even +odd =even
3n^2+odd=even
3n^2=odd-even
3n^2=odd
As 3 is odd =>Hence n must be also to make 3n^2 even.
Thus n is even

Hence this statement is sufficient.

Second way(The more reasonably very quick) => Use hit and trial.
Since n is an integer => It can be either even or odd.
If n is odd => n^2+(n+1)^2+(n+2)^2 => odd+(odd+odd)^2 +(odd+even)^2 => odd+even+odd => even
Hence n can be odd

If n is even => n^2+(n+1)^2+(n+2)^2 => even+odd+even => odd
REJECTED.
Hence n must be always odd.

Sufficient

Third way=> Power does not affect the even/odd nature of any number.
So if n is odd => n^2 is odd
if n is even => n^2 is even too.

n^2+(n+1)^2+(n+2)^2 = even
n+n+1+n+2=> even
3n+3=even
3n => even-odd=> odd
Hence n must be odd.

The same way we can get Statement 2 => Sufficient.

Does this make sense ?

Regards
Stone Cold

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Re: Is the positive integer n odd?  [#permalink]

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01 Jul 2016, 08:13
D
given x is positive integer
only odd integer(n) satisfy both equations
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Re: Is the positive integer n odd?  [#permalink]

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22 Aug 2016, 08:37
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Nice Question
Here we can use the property that POWER does not effect the even or odd nature of any integer
Hence statement 1=> 3n+odd=even => n is odd=> suff
Statement 2=> -n-5=even=> n=even-odd=> odd => suff

Smash that D
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Re: Is the positive integer n odd?  [#permalink]

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22 Aug 2016, 08:50

Statement 1 :- N can only be odd.

as for the statement to be even either all the numbers have to even or there should be a pair of odd numbers i.e.

E+E+E = E
or O+E+O = E

Now as they are consecutive numbers first scenario is not possible. Hence second scenario is correct. Therefore N is odd.

Statement 2 :- Same explanation. Addition or subtraction dont make a difference in even and odd rule.
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Re: Is the positive integer n odd?  [#permalink]

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22 Apr 2017, 08:32
stonecold wrote:
Nice Question
Here we can use the property that POWER does not effect the even or odd nature of any integer
Hence statement 1=> 3n+odd=even => n is odd=> suff
Statement 2=> -n-5=even=> n=even-odd=> odd => suff

Smash that D

stonecold - I am quite a bit lost on this question. Could you just explain a lil bit more how you get to "3n + odd = even" / "-n-5=even"?

p
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Re: Is the positive integer n odd?  [#permalink]

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06 Aug 2017, 23:30
Hi stonecold Bunuel

I understood your explanation but can you suggest flaw in my approach:

I started with : Assuming n is even if st 1 is satisfied:
If n is even, then n+1 is odd and n+2 is even
so we have:
LHS=e+o+e = o ; whereeas RHS is even. Where did I do wrong?
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Re: Is the positive integer n odd?  [#permalink]

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06 Aug 2017, 23:51
1
1
Stmnt 1)
If n is odd
O+E+O = E satisifies

If n is even
E+O+E = O does not satisfy
Hence N is odd

Stmnt 2)
If N is Odd
O-E-O = E satisfies

If N is even
E-O-E = O does not satisfy

Hence answr D
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Re: Is the positive integer n odd?  [#permalink]

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07 Aug 2017, 00:00
hi srikanth9502

thanks for chipping in.
You mean to say that we are looking for unique property of n as even or odd.
Since n is odd satisfies both hence n is odd
Since n is even does not satisfy both hence n is even is not correct
Since I got two answers for n as odd / even for st 1 i rejected it.
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Re: Is the positive integer n odd?  [#permalink]

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16 Oct 2018, 13:42
AbdurRakib wrote:
Is the positive integer n odd?

(1) $$n^2+(n+1)^2+(n+2)^2$$ is even
(2) $$n^2-(n+1)^2-(n+2)^2$$ is even

OG Q 2017 New Question(Book Question: 194)

Because powers don't affect whether the integer is odd or even.

it can be rewritten as n + n +1 + n + 2 = 3n + 3. Only way to get that as a even is if n is odd.

Sufficient.

Apply the same for statement 2

n - n - 1 -n - 2 = -n - 3 is even. the only way for it to be even is if n is odd.

Sufficient.

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Is the positive integer n odd?  [#permalink]

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20 Oct 2018, 05:28
hi srikanth9502

thanks for chipping in.
You mean to say that we are looking for unique property of n as even or odd.
Since n is odd satisfies both hence n is odd
Since n is even does not satisfy both hence n is even is not correct
Since I got two answers for n as odd / even for st 1 i rejected it.

What is "both" that u are referring to?

Question - Is n odd?
If we answer a YES or NO for that then it is sufficient.

Statement 1)
For N to satisfy statement one.
N has to be odd.
So it answers our main question. Yes N is odd.

Statement 2)
For N to satisfy statement two.
N has to be odd here too.
So again it answers our main question. Yes N is odd.

Note - You dont even have to check for statement 2 because all the EVEN and ODD properties for addition are same as that for subtraction.
Example:
2 + 1 = 3 ---------- EVEN + ODD = ODD
2 - 1 = 1 ---------- EVEN - ODD = ODD

Hope it helps.
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Re: Is the positive integer n odd?  [#permalink]

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16 Oct 2019, 05:41
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Re: Is the positive integer n odd?   [#permalink] 16 Oct 2019, 05:41
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