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# If n is an integer, is the prime number y equal to 5 ?

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Math Expert
Joined: 02 Sep 2009
Posts: 46319
If n is an integer, is the prime number y equal to 5 ? [#permalink]

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19 Jan 2018, 11:01
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Difficulty:

55% (hard)

Question Stats:

68% (02:07) correct 32% (01:41) wrong based on 81 sessions

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If n is an integer, is the prime number y equal to 5 ?

(1) y = n^2 + 1
(2) y = n^3 - 3

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Joined: 25 Feb 2013
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If n is an integer, is the prime number y equal to 5 ? [#permalink]

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19 Jan 2018, 20:20
Bunuel wrote:
If n is an integer, is the prime number y equal to 5 ?

(1) y = n^2 + 1
(2) y = n^3 - 3

Statement 1: if $$n=2$$ then $$y=5$$ but if $$n=1$$, then $$y=2$$. Insufficient

Statement 2: if $$n=2$$, then $$y=5$$ but if $$n=4$$, then $$y=61$$. Insufficient

Combining 1 & 2: we have $$n^2+1=n^3-3$$

$$=>n^3-n^2=4$$

$$=>n^2(n-1)=4 =>n^2(n-1)=2^2*1$$

therefore $$n=2$$, hence $$y=5$$. Sufficient

Option C
Math Expert
Joined: 02 Aug 2009
Posts: 5938
Re: If n is an integer, is the prime number y equal to 5 ? [#permalink]

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19 Jan 2018, 20:53
1
Bunuel wrote:
If n is an integer, is the prime number y equal to 5 ?

(1) y = n^2 + 1
(2) y = n^3 - 3
....

statements alone are clearly insuff..

(1) $$y = n^2 + 1$$
when n is 2, y is 5
n is 4, y is 17
n is 1, y is 2
insuff

(2) $$y = n^3 - 3$$
when n is 2, y is 5
n is 4, y is 61
insuff

combined..
$$n^2+1 = n^3 - 3........n^3-n^2=4$$
clearly n^3>n^2, so n has to be positive..
n>4 will make the difference much greater than 4. so try 1,2,3
$$n=1, n^3-n^2=0$$..
$$n=2, n^3-n^2=4$$..
$$n=3, n^3-n^2=23$$..
any number greater will further increase the difference
ans yes
sufficient

C
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Re: If n is an integer, is the prime number y equal to 5 ? [#permalink]

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24 May 2018, 12:14
Bunuel wrote:
If n is an integer, is the prime number y equal to 5 ?

(1) y = n^2 + 1
(2) y = n^3 - 3

(1) n = 2 ---> y = 5; n = 1 ---> y = 2 -----> insuff
(2) n = 2 ---> y=5; n = 4 ---> y = 61 -----> insuff

(1) + (2) n^3 - n^2 - 4 = 0, we see, that of the roots is 2. So let's divide this expression by (n-2)
We get n^3 - n^2 - 4 = (n-2)(n^2 + n + 2)
n^2 + n + 2 = 0 - does not have real roots.
So the only one is n = 2 ---> y = 5 - suff
Re: If n is an integer, is the prime number y equal to 5 ?   [#permalink] 24 May 2018, 12:14
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