If x is a prime number, what is the value of x?

Question

If x is a prime number, what is the value of x?
 
(1) 2x + 2 is the cube of a positive integer.
 
(2) The average of any x consecutive integers is an integer.

Bunuel · Accepted Answer

(1) 2x + 2 is the cube of a positive integer --> 2x+2=n^3 --> n even.

n=2 --> 2x+2=n^3=8 --> x=3=prime;
n=4 --> 2x+2=n^3=4^3 --> x=31=prime;
n=6 --> 2x+2=n^3=6^3 --> x=107=prime.

Multiple choices for x. not sufficient.

(2) The average of any x consecutive integers is an integer. This statement is basically tells us that x is odd, as only the average of odd consecutive numbers are an integer (the average of even number of consecutive integers is a fraction: integer/2). Hence x can be any prime but 2. Not sufficient.

(1)+(2) x can be odd primes, for example 31 or 107, hence insufficient.

Answer: E.

atish · Answer

Got E with a similar approach.
Only tip is for statement 1, instead of prime numbers which satisfy 2*x+2 is a perfect cube, find perfect cubes which when subtracted by 2 and divided by 2 yields a prime number.

xcusemeplz2009 · Answer

IMO E is correct
s1) x = 2 , 31, 107 will give a perfect cube and all are prime so insuff
s2) x can be any prime no. so insuff
togather also no. single value
hence E

mbaiseasy · Answer

Statement 1:
I first listed a few cubes:
1,8,27,64,125,...

x = cube - 2 / 2
x = 8 - 2/2 = 3 (prime)
x = 27 -2 /2 is not an integer (skip)
x = 64 - 2/2 = 31 (prime)

INSUFFICIENT!

Statement 2: A(x) consecutive numbers is an integer if odd (A RULE) and not an integer if EVEN
This is just saying x is odd...
INSUFFICIENT!

Together: 31 and 3 are both ODD
INSUFFICIENT!

Answer: E

mau5 · Answer

From F.S 1, we know that  2(x+1) = t^3 , where t is a positive integer. Two conditions are to be met:

1.x should be a prime number.
2.(x+1) should at-least be equal to  2^2 , or have it as a factor.

Now, for  x+1 = 2^2 , we get x = 4-1 = 3. So , we get x = 3(prime)

Again, to make 2(x+1) as a perfect cube, the next value of (x+1) should be  2^2*2^3 	o x+1 = 32 	o x = 31 (prime). Thus x = 31.

Two values of x 	o Insufficient.

From F.S 2, for a set of consecutive integers, the mean is always the middle term, which is unique and an integer. Thus, for any x, which is odd, the mean of the given series would be a unique integer. Thus, x = odd. Clearly Insufficient.

Both the statements taken together, x can be 3 or x can be 31. Insufficient.

E.

agourav · Answer

(1) INSUFFICIENT: Start by listing the cubes of some positive integers: 1, 8, 27, 64, 125. If we set each of these equal to 2x + 2, we see that we can find more than one value for x which is prime. For example x = 3 yields 2x + 2 = 8 and x = 31 yields 2x + 2 = 64. With at least two possible values for x, the statement is insufficient. 
 
(2) INSUFFICIENT: In a set of consecutive integers, the mean is always equal to the median. When there are an odd number of members in a consecutive set, the mean/median will be a member of the set and thus an integer (e.g. 5,6,7,8,9; mean/median = 7). In contrast when there are an even number of members in the set, the mean/median will NOT be a member of the set and thus NOT an integer (e.g. 5,6,7,8; mean/median = 6.5). Statement (2) tells us that we are dealing with an integer mean; therefore x, the number of members in the set, must be odd. This is not sufficient to give us a specific value for the prime number x.
 
(1) AND (2) INSUFFICIENT: The two x values that we came up with for statement (1) also satisfy the conditions of statement (2). 
 
The correct answer is E.

Sammyh21 · Answer

Statement 1:
Try x=2, works. Potentially the right answer, but x= another prime might also satisfy the equation. Move on.

Statement 2:
X is a prime that is not even. Not sufficient.

1 & 2 together:
Since both statements can't contradict each other, it means that x=2 can't be the answer. Both statements don't narrow x down to one value. Answer is E

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GMATinsight · Answer

Let me mention it here...

YOU NEVER ASSUME THAT STATEMENT IS  NOT SUFFICIENT  BASED ON HUNCH LIKE YOU DID IN STATEMENT 1. Also x=2 doesn't work

You must be able to prove that it's not sufficient in order to improve accuracy therefore the first statement should be considered in the following manner

Statement 1: 2x + 2 is the cube of a positive integer

2x+2 = 2(x+1) for this to be a perfect cube (x+1) must be even i.e. x must be odd

at x=3, 2x+2 = 8 One possibility
another Even perfect cube is 64
64 = 2(x+1) => x=31

there are two or more possibilities of x= 3, 31 etc.
Inconsistent Values therefore,
NOT SUFFICIENT

Statement 2: Average of X consecutive integers is even

True for all Odd values of x therefore
NOT SUFFICIENT

Combining the two statements

x can still be 3 or 31 etc
so, NOT SUFFICIENT

Answer: Option  E

AmritaSarkar89 · Answer

Aha question as Mike from Magoosh will say

well rather than hitting and trying with multiple prime numbers we can opt for a simpler process. first sort out the cube roots and see if any equation is satisfying it with prime number. 2^3=8 2(x+1)=8 yes prime 3 next 3^3= 27 No primes satify this equation next 4^3=64 any prime satisfying it ? YES 31 DONE STOP PROCEEDING Statement 1 is Not sufficient Statement 2: well any odd number will satify this statement and we can have multiple answers such as 3 and 31 which are primes. Combining 1& 2 we still have multiple options Thus E

yezz · Answer

from 1

x = y^3/2 - 1 thus y has to be even , y = 2 thus x could be 7 , or y = 4 and thus x = 3... insuff

from 2

x is odd prime ( 3,5,7,...etc)

both

x could be 3 or 7... insuff

E

edgehead · Answer

C as both the statements on their own are not sufficient ,when clubbed result in x=3

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stonecold · Answer

Great Question.
We are told that x is a prime number and asked about its value.

Statement 1 ->
2x+2=t^3
for any positive integer t.
Lets use some test cases =>
2x+2=1
x=1/2=> Not allowed
2x+2=8
x=3=> Allowed
2x+2=27
x=25/2=> Not allowed
2x+2=64
x=31=>Allowed

Hence 3 and 31 are both acceptable values => Not sufficient.

Statement 2 ->
Remember for set of consecutive integers is an AP series.
Mean can be of the form p if x is odd (p is an integer )
Mean can be of the form p.5 if x is even (p is an integer)
 RULE->Sum of n consecutive integers is always divisible by n for n being odd and never divisible by n for n being even. 
Hence this statement tells us that x is odd.
But their are ∞ odd primes => Not sufficient.
Combing them x=3 and x=31 are both acceptable values.
Hence not sufficient .

Hence E.

aserghe1 · Answer

Wanted to contribute my approach for Statement 1:

2x+2=2(x+1)  is the cube of a positive integer.

We can have  x+1=2^2 , which gives us x=3.
Or we can have  x+1=2^2*3^3 , which gives us x=107.
Not sufficient, since both 3 and 107 are primes.

MBAaustralia · Answer

I'm horrible with prime questions. Can anyone recommend a good resource that covers this area?

Posted from my mobile device

Bunuel · Answer

2. Properties of Integers  
     Theory   
 Math: Number Theory 
 Number Properties: Tips and hints 
 Advanced Number Properties on GMAT (All 5 parts) 
 Do you make these 3 mistakes in GMAT Even-Odd Questions?  
 Everything about Factorials on the GMAT 
 Solving Complex Problems Using Number Line 
     Questions   
 Tough New Set: Number Properties Set 
 Power of a Number in a Factorial Problems 
 Number Properties Ability Quiz 
 Special Numbers and Sequences Problems 
 The E-GMAT Number Properties Knockout: 10 Great 700+ Ques for you! 
 Trailing Zeros Problems 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

MBAaustralia · Answer

Yes, very helpful as usual Bunuel!

bumpbot · Answer

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If x is a prime number, what is the value of x?

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