If x and y are positive integers, is x even? (1) y^x is a prime number

Question

GMAT CLUB'S FRESH QUESTION

If x and y are positive integers, is x even?

(1) y^x is a prime number
(2) x*y is a prime number

M36-30

SSM700plus · Answer

Solution:-

Statement 1 :-  y^x  is a prime number

If any number is raised to the power. The result will not be prime numbers.

For Statement 1 to be prime number.   X has to be ONE  .

Is X even. Ans is NO.

Hence it is sufficcient  .

Statement 2 :- x*y is a prime number

Product of two number is prime only if any one of X & Y is ONE.

But we cannot determine, If X = 1 or Y =1.

For example -
1) x =2 & y=1. X * Y = 2 * 1 = 2. is a prime number. And X is even.

2) x =1 & y=3. X * Y = 3 * 1 = 3. is a prime number. And X is odd.

No definite ans

Hence statement 2 is insufficient.

Ans - A  .

PLUTO · Answer

what if I assume that y=1 and vary x = 2,3,.... For all those options ST1 not sufficient as X can be odd or even. Please correct me.

PKN · Answer

Hi  ANGHOS82 ,

Notes:-

1) An odd number can be prime. (Example 1, 3,7,11 etc)
2) An even number can be prime. (Example only 2)
3)  1 is not prime. 
4) 1 is odd.
5) 1 raised to the power any positive integer is equal to 1.

As per your picked numbers,

y=1, x=2; so  y^x=1^2=1  , which is not a prime number, violating statement 1.

So, we can't pick y=1.

Hope it helps.

NYCMBA2010 · Answer

Hi, I tested x=0, to get that raising anything to 0 will result in a prime number, 1. Raising a prime number to 1, will result (obviously) to that same prime number. Hence, x could be either 0 or 1.

I made a silly mistake since it clearly stated that x and y are positive integers, so that would eliminate 0 as an option and make statement 1 sufficient (always odd). Just writing down my question helped to spot where I went wrong.

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

Archit3110 · Answer

#1:
3^1 or 2^1 prime no so x has to be 1 only sufficient

#2: x*y= prime x can be either even or odd in case of 2*1 so in sufficient

IMOA

Ypsychotic · Answer

Statement 1) a prime no has only 2 factors, 1 and itself. So, if y^x is prime. No of factors=x+1, x+1=2, x=1, x is odd. Sufficient.
Statement 2) x can 1 or the prime number. If xy=2, x can be 1 or 2. Either odd or even. Insufficient.
Hence, Option A is correct.

Bunuel · Answer

Official Solution:

If  x  and  y  are positive integers, is  x  even? 
 
(1)  y^x  is a prime number
 
The only way the above to be true is when  y=prime  and  x=1=odd . If  y 
eq {prime} , then no positive integer in positive integer power can be prime. And if  x
eq 1 , then  prime^x  won't be a prime. Sufficient.
 
(2)  x*y  is a prime number 
 
A prime number has only two positive factors: 1 and itself. Can  x  be odd? Yes, if  y=prime  and  x=1 . Can  x  be even? Yes, if  y=1  and  x=2 . Not sufficient.

Answer: A

If x and y are positive integers, is x even? (1) y^x is a prime number

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