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

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AbeNeedsAnswers Master | Next Rank: 500 Posts Default Avatar
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OG If x is a positive integer, then is x prime?

Post Thu Aug 24, 2017 8:38 pm
If x is a positive integer, then is x prime?

(1) 3x + 1 is prime
(2) 5x + 1 is prime

E

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Post Fri Dec 08, 2017 7:38 am
AbeNeedsAnswers wrote:
If x is a positive integer, then is x prime?

(1) 3x + 1 is prime
(2) 5x + 1 is prime
We are given that x is a positive integer and must determine whether x is prime.

Statement One Alone:

3x + 1 is prime.

Using the information in statement one, x does not necessarily have to be prime. For instance, if x = 2, then 3x + 1 = 7 is prime, or if x = 4, then 3x + 1 = 13 is prime. In the former case, x = 2 is prime; however, in the latter case, x = 4 is not prime. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

5x + 1 is prime.

Using the information in statement two, x does not necessarily have to be prime. For instance, if x = 2, then 5x + 1 = 11 is prime, or if x = 6, then 5x + 1 = 31 is prime. In the former case, x = 2 is prime; however, in the latter case, x = 6 is not prime. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using the information from statements one and two, we still cannot determine whether x is prime. For instance, if x = 2, then both 3x +1 = 7 and 5x + 1 = 11 are prime, or if x = 12, then both 3x + 1 = 37 and 5x + 1 = 61 are prime. In the former case, x = 2 is prime; however, in the latter case, x = 12 is not prime.

Answer: E

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Post Fri Aug 25, 2017 12:34 am
AbeNeedsAnswers wrote:
If x is a positive integer, then is x prime?

(1) 3x + 1 is prime
(2) 5x + 1 is prime

E
Given: x is a positive integer
To find out: is x prime?

Statement 1: 3x + 1 is prime

Since x is a positive integer, the minimum value of x = 1, thus the minimum value of 3x + 1 = 3*1 + 1 = 4, but '4' is non-prime, thus x = 1 is not a valid value.

Let's try with x = 2, then 3x = 1 = 3*2 + 1 = 7, a prime number. So, if x = 2, then x is prime.

We know that all the prime numbers except 2 are odd, thus, 3x + 1 = Odd.

3x + 1 = Odd
3x = Odd - 1 = Odd - Odd = Even
x = Even / 3 = Even / Odd = Even (since x is a positve integer);

Note that if an even integer is completely divisible by an odd integer, the quotient is even.

So, we have x = Even

If x = 2, the answer is Yes; however, if x ≠ 2, the answer is No. Insufficient.

Statement 2: 5x + 1 is prime

At x = 2, we see that 5x + 1 = 5*2 + 1 = 11, a prime number, so if x = 2, the answer is yes.

As discussed in Statement 1, we know that all the prime numbers except 2 are odd, thus, 5x + 1 = Odd.

5x + 1 = Odd
5x = Odd - 1 = Odd - Odd = Even
x = Even / 5 = Even / Odd = Even (since x is a positve integer);

So, we have x = Even

If x = 2, the answer is Yes; however, if x ≠ 2, the answer is No. Insufficient.

Statement 1 & 2:

x = 2, implying x is prime, and x = other even integers are applicable in both the statements, thus there is no unique answer. Insufficient.

The correct answer: E

Hope this helps!

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