Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n

Question

Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n is an integer. Source:Nova I'm clueless about this one.

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Bunuel · Answer

Is n^2 + 2n an integer?

(1) n^3 + 3n is an integer --> if  n=1  then the answer is YES. But if  n^3+3n=1  then  n=irrational \ number\approx{0.32} , for which  n^2 + 2n
eq{integer} . So, when  n  is an integer then the answer is YES but if  n  is some irrational number then the answer is NO. Not sufficient.

(2) 2n is an integer -->  n  is either an integer (answer YES) or  \frac{integer}{2}  (answer NO). Not sufficient.

(1)+(2) From above  n  can only be an integer, which makes  n^2 + 2n  equal to an integer. Sufficient.

Answer: C.

P.S. Not a good question, not a good source.

conty911 · Answer

Thanks for the explanation.After solving 50-60 questions, i also felt that the NOVA book, is not that good as explantions for some of the answers are confusing, and further it demoralises me when, I fail to understand/solve the problem.

pavanpuneet · Answer

It is clear that statement a and b alone are not sufficient, however when we combine 1 and 2..

=> n3 + 3n = I; n^3 + 2n +n = I; n^3+n = I <given that 2n is an integer> = n3 + n can be integer only if n is integer.. therefore C.

vikram4689 · Answer

how did you arrive at above statement

Why ? bunnel calc. n for n^3 + 3n =1 , so in a similar way(i dont know how he calculated, may be he can tell) n^3+n=integer can be calculated

Bunuel · Answer

From (1)  n  is either an integer or an irrational number;
From (2)  n  is either an integer or an  \frac{integer}{2} ;

For (1)+(2)  n  can only be an integer.

Hope it's clear.

vikram4689 · Answer

how did you find that solution of n^3+3n=1 is an irrational no.

Bunuel · Answer

Solution of n^3+3n=1 is not an integer or a fraction (try to prove it) so it must be an irrational number.

vikram4689 · Answer

on same lines, can i say that n^3+n=integer is also irrational

Bunuel · Answer

Please read the solutions carefully.

If n is an integer then n^3+n is always an integer. But n^3+n=integer does not necessarily means that n is an integer.

vikram4689 · Answer

Agreed, i am referring to above poster who said   n3 + n can be integer only if n is integer  but i was thinking that n can be irrational as well. Now how do i arrive at a conclusion.

Bunuel · Answer

I don't understand your question.

Again, (1) says: n^3 + 3n is an integer. n^3+n=integer can occur when n=integer (for example n^3+n=2 --> n=1) as well as when n=irrational number (for example n^3+n=3 --> n=irrational number).

vikram4689 · Answer

i totally understand the solution you have provided. i was reading below solution given by some other poster and i think he is wrong when he mentioned  n3 + n can be integer only if n is integer..  . Although he mentions answer correctly

vibhav · Answer

between c & e i picked e. also but now C makes sense. in any case the OA provided is just wrong.

stne · Answer

when they say n^3+3n is an integer I thought n has to be an integer

The other argument that n^3+3n = integer does not necessarily mean n is an integer , may require some more examples.

for example n^3+3n = 1 solving these we get n = 0.32( approx) but on putting n = 0.32 in eqn n^3+3n we  do not  get 1. if n = 0.32 then n^3+3n = 0.992768 ( not an integer)

Is there an exact non integer value of n so that on putting the value of n is the equation n^3+3n we get an integer?

I went with A on this one.

Bunuel · Answer

The reason you don't get exactly 1 is because you do not plug exact value of n. The exact solution of n for n^3+3n = 1 is  n = \sqrt {\frac{1}{2} (1+\sqrt5)}-\sqrt {\frac{2}{1+\sqrt5}}

stne · Answer

Solving the above expression

Based on how you calculate it I got n = 0.322185354 and another values as n= 0.322185355

putting the first value of n in n^3+3n we get n^3+3n = 0.999999998 still not equal to 1

putting the second value of n in n^3+3n we get 1.000000001 still not equal to 1

Using a scientific calculator I tried to get to the bottom of the matter.But no matter how hard I try I cannot get a non integer value of n which leads to integer value of the expression n^3+3n.

Bunuel · Answer

Then you just have to believe...

Bunuel · Answer

Here is a link to WolframAlpha site showing the exact result: https://www.wolframalpha.com/input/?i=%2 ... 5E1%2F3%29 and the screenshot: 2014-02-13_1719.png

stne · Answer

Thank you , certainly not an easy one.+1

Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n

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Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n

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