Number Properties: Prime Number

[Gmater-1]

If i am missing some vital point here

If N = (x^a)*(y^b), where x and y are prime numbers, and a and b are positive intergers. Is N^1/2 an integer

1. a+b is even
2. a*b is even



stmt 1

2^2 + 3^2 = 13 and sqrt 13 is not integer
2^4 + 3^2 = 25 sqrt 25 is integer
Not sufficient

Stmt 2
2^1 + 3^2 = 11 and sqrt 11 is not integer
2^4 + 3^2 = 25 and sqrt 25 is integer

Not suff

Combined
2^4 + 3^2 = 25 and sqrt 25 is integer
2^2 + 3^2 = 13 and sqrt 13 is not integer

Not suff Hence answer should be E but Official Answer is

SPOILER: c

am i missing something here


thanks

[manish8109]

Let x=2 & y=3

Statement 1:

a+b is even
N= (x^a)*(y^b)

a=2, b=2 ------ Both even
then N^1/2 will always be an integer
N= (2^2)*(3^2)= 36 => N^1/2 = +/- 6 -----Integer

a=3, b=1----------> both odd
then N^1/2 will never be an integer
N= 2^3 * 3^1 = 24 => N^1/2 ------Not an Integer

Insufficient

Statement 2:

a=1 & b=2

N= 2^1 * 3^2 = 18 => N^1/2--------Not an Integer

a=2 & b=2

N= 2^2 * 3^2 = 36=> N^1/2 -------Integer

Combining both

Either of a or b one should always be even to make a*b even & gor a+b to be even then b should always be even.

i.e. N= x^2 * y^b => N^1/2----------Would always be an integer for even a & b.

Ans(C)