n^2 is a positive integer

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 amFollowed by: 80 members; Location: India; Thanked: 267 times; GMAT Score:760

by sanju09 » Wed Mar 04, 2009 3:54 am

If n is an integer and if n^2 is a positive integer, which of the following must also be a positive integer?

(A) n^2 + n
(B) 2 n^2 – n
(C) n^2 – n^3
(D) n^3 + n
(E) 2 n^3 + n

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Source: Beat The GMAT — Problem Solving | Wed Mar 04, 2009 3:54 am

DanaJ: Site Admin; Posts: 2567; Joined: Thu Jan 01, 2009 10:05 amFollowed by: 550 members; Thanked: 712 times; GMAT Score:770

by DanaJ » Wed Mar 04, 2009 5:57 am

You get that, since n^2 is a positive integer, n will be any integer except for 0.
Eliminate a few options quickly, particularly D and E, by plugging in any negative number, like for instance n = -2.

In order to asses the other stmts, we need to remember a property of the quadratic equation. Take the general case of a quadratic equation:
ax^2 + bx + c
When a is positive, the above equation will be negative only between the roots, while it will be positive for the rest of the bunch.

Now take A. n^2 + n = n(n + 1) - this is a quadratic equation, which we know is negative only between the roots, 0 and -1. But since there are no integers between -1 and 0, the equation is never negative. However, take n = -1 and you get that n^2 + n = 0, which is of course, not positive.

Time for B. 2n^2 - n = n(2n - 1) - again, negative between the roots, with roots 1/2 and 0. Since there are no integers between 0 and 1/2 and 0 has been eliminated at the beginning, this is what we're looking for. So my answer is B

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El Cucu: Master | Next Rank: 500 Posts; Posts: 236; Joined: Fri Nov 28, 2008 9:16 am; Thanked: 9 times

Re: n^2 is a positive integer

by El Cucu » Wed Mar 04, 2009 7:22 am

sanju09 wrote:If n is an integer and if n^2 is a positive integer, which of the following must also be a positive integer?

(A) n^2 + n
(B) 2 n^2 – n
(C) n^2 – n^3
(D) n^3 + n
(E) 2 n^3 + n

Plug in -2, -1, 1 & 2.(as n is an integer)

Only B) satisfies the condition.

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marcusking: Master | Next Rank: 500 Posts; Posts: 106; Joined: Wed Feb 04, 2009 6:09 am; Location: Louisville, KY; Thanked: 8 times

Re: n^2 is a positive integer

by marcusking » Wed Mar 04, 2009 7:25 am

sanju09 wrote:If n is an integer and if n^2 is a positive integer, which of the following must also be a positive integer?

(A) n^2 + n
(B) 2 n^2 – n
(C) n^2 – n^3
(D) n^3 + n
(E) 2 n^3 + n

I try to prove the equation wrong to do so I try the smallest negative integer n=-1 to try and make the answers negative
A)-1^2 = 1 + -1 = 0 Not a positive integer
B) 2 * 1 - (-1) = 2 + 1 = 3 (Works)

Now try B with the smallest positive integer +1
2 * 1 - 1 = 1 (Works)

It has satisfied both a positive and negative integer, if I go further in either direction the 2*n^2 will only become a larger positive number

ANS. B.

Please start posting the OA.

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 amFollowed by: 80 members; Location: India; Thanked: 267 times; GMAT Score:760

by sanju09 » Thu Mar 05, 2009 2:23 am

If n is an integer and if n^2 is a positive integer, then 2n^2 – n = 2n^2 + (–n) is the sum of two integers, and must therefore be an integer. Since n^2 is a positive integer, it follows that |n| ≥ 1. The integer 2n^2 – n will be positive if 2n^2 > n. Since |n| ≥ 1, it follows that n^2 > n, and so 2n^2 > n. Therefore, if n^2 is a positive integer, 2n^2 – n must also be a positive integer.

If n^2 is a positive integer, then the expressions in the other choices must be integers, but they need not be positive integers:

For n^2 + n, if n = –1, then n^2 + n = (–1)^2 + (–1) = 1 – 1 = 0, which is not positive.

For n^2 – n^3, if n = 2, then n^2 – n^3 = 2^2 – 2^3 = 4 – 8 = –4, which is not positive.

For n^3 + n, if n = –2, then n^3 + n = (–2)^3 + (–2) = –8 – 2 = –10, which is not positive.

For 2n^3 + n, if n = –2, then 2n^3 + n = 2(–2)^3 + (–2) = –16 – 2 = –18, which is not positive.

Therefore, the correct answer is [spoiler](B)[/spoiler].

The mind is everything. What you think you become. -Lord Buddha

Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com

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