If x is a positive integer, is x a prime integer?

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If x is a positive integer, is x a prime integer?

(1) x+1 is a prime number.
(2) x-5 is a prime number.

OA C

Source: GMAT Prep

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BTGmoderatorDC wrote:If x is a positive integer, is x a prime integer?

(1) x+1 is a prime number.
(2) x-5 is a prime number.
Source: GMAT Prep
$$x \geqslant 1\,\,\operatorname{int} $$
$$x\,\,\mathop = \limits^? \,\,{\text{prime}}$$

$$\left( 1 \right)\,\,x + 1\,\,{\rm{prime}}\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,x = 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,x = 2\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.$$

$$\left( 2 \right)\,\,x - 5\,\,{\rm{prime}}\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,x = 7\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,x = 8\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,\left\{ \matrix{
\,\left( 2 \right)\,\,\, \Rightarrow \,\,\,x - 5 \ge 2\,\,\, \Rightarrow \,\,\,x \ge 7\,\,\,\left( * \right) \hfill \cr
\,\left( 1 \right)\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,x + 1\,\,{\rm{odd}}\,\, \ge \,\,{\rm{9}}\,\,\,\,\, \Rightarrow \,\,\,\,\,x \ge 8\,\,{\rm{even}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$

In words: x+1 is prime and greater than 7, hence x+1 is odd. Therefore x is even and greater than 6, hence x is not prime!


The correct answer is (C).



This solution follows the notations and rationale taught in the GMATH method.

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Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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by Jay@ManhattanReview » Sun Dec 02, 2018 4:57 am

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BTGmoderatorDC wrote:If x is a positive integer, is x a prime integer?

(1) x + 1 is a prime number.
(2) x - 5 is a prime number.

OA C

Source: GMAT Prep
Given: x is a positive integer.

Question: Is x a prime integer?

Let's take each statement one by one.

(1) x + 1 is a prime number.

Case 1: Say x = 1 (non-prime), then x + 1 = 2 is a prime number. The answer is No.
Case 2: Say x = 2 (Prime), then x + 1 = 3 is a prime number. The answer is Yes.

No unique answer. Insufficient.

(2) x - 5 is a prime number.

Case 1: Say x = 8 (Non-prime), then x - 5 = 3 is a prime number. The answer is No.
Case 2: Say x = 7 (Prime), then x - 5 = 2 is a prime number. The answer is Yes.

No unique answer. Insufficient.

(1) and (2) together

Given that x + 1 and x - 5 are prime numbers, we see that the difference between them is 6 [= x + 1 - (x - 5)]. Since 6 is even thus, the two prime numbers x + 1 and x - 5 are odd. Since x +1 is odd, we have x is even. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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