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Math Expert V
Joined: 02 Sep 2009
Posts: 64090
If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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75 00:00

Difficulty:   65% (hard)

Question Stats: 60% (01:57) correct 40% (01:56) wrong based on 1742 sessions

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

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

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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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23
There is one variable (x) in the original condition. So we need 1 equation. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that D is the correct answer.
The condition 1) has x=2 yes, x=6 no not suffi.
The condition 2) has x=2 yes, x=6 no not suffi.
Since the condition 1) and the condition 2) have x=2 yes, x=6 no not suffi, the correct answer is E.

- For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
Attachments variable approach's answer probability.jpg [ 219.74 KiB | Viewed 23483 times ]

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If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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1
A quick trick: plug in only even numbers in $$x$$.
Remember that with the exception of 2, prime numbers are odd.

For the equation $$3x + 1$$ and $$5x + 1$$ to have an odd prime result, we need to plug in even numbers 2, 4, 6 to $$x$$.

Because combining both A and B gives us $$x = 2$$ (prime) and $$x = 6$$ (not prime), our answer is E.
##### General Discussion
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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Bunuel wrote:
If x is a positive integer, then is x prime?

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

x = +ve integer
(1) 3x + 1 is prime
if 3x+1 = 7 = prime
=> x= 2 , which is prime

if 3x+1 = 19 = prime
=> x = 6 , which is not a prime
Not sufficient

(2) 5x + 1 is prime
if 5x + 1 = 11 = prime
=> x = 2 , which is prime

if 5x + 1 = 31 = prime
=> x = 6 , which is not a prime
Not sufficient

Combining 1 and 2 , we get

Still Not sufficient
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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1
Bunuel wrote:
If x is a positive integer, then is x prime?

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

Statement 1:-

If x is 2, 4 , 6.... then 3x + 1 is prime

if x is 1, 3, 5.... then 3x + 1 is not prime

Not sufficient as x may or may not be prime.

Statement 2:-

If x is 1, 3, 5... then 5x + 1 is not prime

if x is 2, 6... then 5x + 1 is prime.

Not sufficient as x may or may not be prime.

Both statements combined together do not give a unique solution. Hence the answer is E
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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2
Bunuel wrote:
If x is a positive integer, then is x prime?

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

primes > 3 are always of the form 6k +/- 1

1) 3x + 1 = 6k+1
=> x = 6k/3 = 2k
2k can be prime (2) or not (4,6,8,...)
insufficient

2) 5x+1 = 6k+1
=> x = 6k/5
here x cannot be a prime

5x+1 = 6k-1
=> x = (6k-2)/5
if k=2, then x=2 (prime)
if k=7, then x=8 (not a prime)
insufficient

(1)+(2)
x=2 or x=8 (more common values might exist but i am not bothering to calculate since this is enough to answer)
insufficient.

Hence, E.
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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1
Bunuel 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.

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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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Plug in prime numbers as a result of the equation
1)
3x+1=7
x=2 (prime)

3x+1=19
x=6 (not prime); hence NS

2)
5x+1=11
x=2 (prime)

5x+1=31
x=6 (not prime) ; hence NS

1) & 2)
x=2 or x= 6 and both equations would still yield a prime number.

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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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Hi Bunuel,

Is there an alternative method to this where we do not have to plug in values at random?

Best-
Amit
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Posts: 64090
Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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Amit989 wrote:
Hi Bunuel,

Is there an alternative method to this where we do not have to plug in values at random?

Best-
Amit

For this particular question you have to test values. There is no purely algebraic way.
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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1
\sqrt{}
Bunuel wrote:
If x is a positive integer, then is x prime?

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

(1) 3x + 1 is prime

Let x =2........3 + 1 =7........Answer is Yes

Let x =6........18 + 1 =19........Answer is No

Insufficient

(2) 5x + 1 is prime

Let x =2........10+ 1 =11........Answer is Yes

Let x =6........30 + 1 =31........Answer is No

Insufficient

Combine 1 & 2

Insufficient

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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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Bunuel wrote:
If x is a positive integer, then is x prime?

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

1) Given: 3x+1 is prime, also since x>=1 , 3x+1 cant be 2, this implies 3x+1 is an odd prime, thus 3x is even,
Thus x is even. Now x>=1, so x can be 2 (prime) and greater than 2(Not prime). Insufficient
2) We can use the similar explanation for 5x+1. Insufficient.

Combining both is also insufficient.
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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1
Bunuel wrote:
If x is a positive integer, then is x prime?

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

Bunuel chetan2u
can u clear this confusion:
3x+1 is prime
so 3x+1=6k+1 or 3x+1=6k-1
x=2k ----(A) or x=2k-(2/3)-----(B)
eqn b can never be prime as it will always be decimal
eqn A can be prime if k=1 or cannot be prime if k=2
so cannot say
My question is
will the answer be common to both A and B which is No
if i prove in any case x can be prime then it will be yes
else no
so we are considering its insufficent
can u please claer this ?
how to interpret answer from both the statemnts
Say it had been and case
then answer will be No ,right?
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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This is an excellent question on the properties of Primes. I’d also use the Units digit concept to solve the question since there is a multiple of 5 mentioned in one of the statements.

‘x’ is a positive integer. This means that x can take any value from the set {1,2,3,4,5……}.

From statement I, we know that 3x + 1 is prime. Since x is a positive integer, we can say that the smallest value we can take for 3x is 3, which is to say that the smallest value for x is 1. Therefore, 3x + 1 cannot be equal to 2 or 3.

If 3x + 1 = 5, 3x = 4. x is not an integer here. If 3x + 1 = 7, 3x = 6 which gives us x = 2. This value of x answers the main question with a YES since 2 is prime.

Any prime number greater than 3 can be written as 6k – 1 or 6k + 1 where k is a positive integer. A careful analysis will tell us that when we equate 3x + 1 to 6k + 1, we get x to be an integer. That’s why we got x = 2 in the example above.

The next two prime numbers that can be written in the form of 6k + 1 are 13 and 19. If 3x + 1 = 13, x = 4; if 3x + 1 = 19, x = 6. 4 and 6 are not prime numbers and hence answer the main question with a NO.

Statement I is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, 5x + 1 is a prime. Again, the smallest prime of this form can be 11. Clearly we are dealing with odd primes. Therefore, 5x has to be even. In other words, 5x is a multiple of 5 with units digit ZERO and so 5x + 1 is a prime with units digit 1. This means that x can be 2 or 6 or 8. We again face a YES NO situation.

Statement II alone is insufficient. Answer option B can be eliminated. Possible answer options are C or E.

Combining the data given in the two statements, we have x = 2 and x = 6 satisfying both statements at the same time. The combination of statements is insufficient as well.

The correct answer option is E.

Hope that helps!
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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Ideally I would test numbers for this but if I were in a real hurry to solve this question, could I argue that prime numbers do not follow any regular pattern and therefore mark E straightaway?
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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Debo1988 wrote:

Ideally I would test numbers for this but if I were in a real hurry to solve this question, could I argue that prime numbers do not follow any regular pattern and therefore mark E straightaway?

Yes, you can but keep in mind that though there is no pattern to the generation of prime numbers, there are some properties that all prime numbers follow. That is, there is no property which tells you which number would be the next prime but once you find a prime, it will follow certain properties e.g. all primes greater than 3 are of the form (6x+1) or (6x-1).
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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Can you please check your e-mail, I js emailed you my query
Kindly guide me

Posted from my mobile device
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Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  [#permalink]

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x=pos int
Is x= prime?
Statement 1: 3x+1=prime
Insufficient
Statement 2: 5x+1=prime
Both statements together:
Again x=2 and x=6 give a yes and no answer
Insufficient
E Re: If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x   [#permalink] 21 Apr 2020, 21:54

# If x is a positive integer, then is x prime? 1) 3x + 1 is prime 2) 5x  