Data Sufficiency

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

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

(2) The average of any x consecutive integers is an integer.

OAE

Hi Needgmat,

When it comes to number theory questions the best approach is to think of using numbers instead of algebra. While using numbers make sure to test for different types of numbers (positives, negatives, fractions and zero).

In this question plugging in numbers is the fastest approach since we have a constraint that x has to be prime. Also keep in mind that this is a value DS problem, so for a statement or both statements to be sufficient they have to give us one unique answer.

Statement 1 : 2x + 2 is a cube of an integer

Now 2x + 2 can be 8, 27, 64, 125, 216..., but since x has to be prime x

2x + 2 = 8 ; x = 3
2x + 2 = 64 ; x = 31
2x + 2 = 216 ; x = 107

We cannot use 2x + 2 as 27 and 125 because in these cases we cannot have x as prime (x will not be an integer)

So x = 3 or 31 or 107. Since this is not one definite answer, statement 1 is insufficient.

Statement 2 : The average of x consecutive integers is an integer

This statement is easy to evaluate as x here can be any prime number like 3, 5, 7, 31..... However there is something you can take back from this statement.

'The average of odd number of consecutive integers will always be an integer' and 'The average of an even number of consecutive numbers will never be an integer'.

So the statement here just tells us that x is odd. Insufficient.

Now combining the two statements also does not yield a definite answer as x can be any prime out of 3, 31, 107.... Insufficient.

Hope this helps!

CrackVerbal Academics Team

When it comes to number theory questions the best approach is to think of using numbers instead of algebra. While using numbers make sure to test for different types of numbers (positives, negatives, fractions and zero).

In this question plugging in numbers is the fastest approach since we have a constraint that x has to be prime. Also keep in mind that this is a value DS problem, so for a statement or both statements to be sufficient they have to give us one unique answer.

Statement 1 : 2x + 2 is a cube of an integer

Now 2x + 2 can be 8, 27, 64, 125, 216..., but since x has to be prime x

2x + 2 = 8 ; x = 3
2x + 2 = 64 ; x = 31
2x + 2 = 216 ; x = 107

We cannot use 2x + 2 as 27 and 125 because in these cases we cannot have x as prime (x will not be an integer)

So x = 3 or 31 or 107. Since this is not one definite answer, statement 1 is insufficient.

Statement 2 : The average of x consecutive integers is an integer

This statement is easy to evaluate as x here can be any prime number like 3, 5, 7, 31..... However there is something you can take back from this statement.

'The average of odd number of consecutive integers will always be an integer' and 'The average of an even number of consecutive numbers will never be an integer'.

So the statement here just tells us that x is odd. Insufficient.

Now combining the two statements also does not yield a definite answer as x can be any prime out of 3, 31, 107.... Insufficient.

Hope this helps!
CrackVerbal Academics Team

Hi CrackVerbal ,

Thank you so much for your explanation. It really helps.

Thanks,

Kavin

given x is a prime number

(1) 2x+2 cube root of an integer
let x=3 =2*3 + 2 = 8(cube root of 2)
let x=31 =2*31 + 2= 64(cube root of 4)
NOT SUFFICIENT

(2) average of consecutive x terms is an integer
average=(x+1)/2
x = 3 , average = 2
x = 31, average = 16
NOT SUFFICIENT

combining
it can be either 3 or 31 or more numbers
NOT SUFFICIENT

SO E