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If p is a positive integer, is 2p + 1 prime?

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by fskilnik@GMATH » Tue Oct 30, 2018 6:40 pm
BTGmoderatorDC wrote:If p is a positive integer, is 2p + 1 prime?

(1) p is prime
(2) units digit of p is not prime

Source: Official Guide
$$p \ge 1\,\,{\mathop{\rm int}} $$
$$2p + 1\,\,\mathop = \limits^? \,\,{\rm{prime}}$$

Let´s go straight to (1+2): a BIFURCATION will guarantee (E) as the correct answer!

$$\left( {1 + 2} \right)\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,{\rm{p = 11}}\,\,\,\, \Rightarrow \,\,\,2p + 1 = 23\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,{\rm{p = 19}}\,\,\,\, \Rightarrow \,\,\,2p + 1 = 39\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$

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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
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by Jay@ManhattanReview » Tue Oct 30, 2018 10:01 pm
BTGmoderatorDC wrote:If p is a positive integer, is 2p + 1 prime?

(1) p is prime
(2) units digit of p is not prime

OA E

Source: Official Guide
Given: p is a positive integer.
Question: Is 2p + 1 prime?

Let's take each statement one by one.

(1) p is prime.

Case 1: Say p = 2, then 2P +1 = 5, a prime. The answer is Yes.
Case 2: Say p = 7, then 2P +1 = 15, not a prime. The answer is No.

No unique answer. Insufficient.

(2) Units digit of p is not prime.

Case 1: Say p = 11, then 2P +1 = 23, a prime. The answer is Yes.
Case 2: Say p = 19, then 2P +1 = 39, not a prime. The answer is No.

No unique answer. Insufficient.

(1) and (2) together

Both the cases discussed in Statement 2 are applicable here too.

No unique answer. Insufficient.

The correct answer: E

Hope this helps!

-Jay
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