If k is a positive integer

[atulmangal]

If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4.
(2) k is divisible by exactly four different prime numbers.

[manpsingh87]

If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4.
(2) k is divisible by exactly four different prime numbers.

Q1)
1) consider k=36 36 is a perfect square and is divisible by 4, also consider k=12; 12 is divisible by 4 but it is not a perfect square hence 1 is insufficient.

2) k=(2^2)*(3^4)*(5^2)*(7^2); now k is a perfect square and is divisible exactly by 4 prime factors, now consider k=2*3*5*7; now also k is divisible by 4 different prime numbers but it is not a perfect square... therefore 2 alone is not sufficient to answer the question..!!

combining 1 and 2,

consider k=(2^2)*(3^4)*(5^2)*(7^2); it is divisible by 4 and is a perfect square, now consider k=(2^2)*3*5*7; it is also divisible by 4 but is not a perfect square hence answer should be E

[rohu27]

there are values either way satisfying both 1 and 2, manpsingh has pointed them out beautifully, so it got to be E.

[atulmangal]

Hey Guys,

OA is E only, even i applied that same technique (@manpsingh87) and then thought to put this question in this forum so that people may share some other techniques also..

[MAAJ]

Same as above, (E) here's what I did:

If k is a positive integer, is k the square of an integer?
k = +integer; k = x²?

(1) k is divisible by 4.
k -> 2,2,...? in its prime factors
If k -> 2,2 then k will be a perfect square.
If k -> 2,2,3 then k will not be a perfect square.
Insufficient

(2) k is divisible by exactly four different prime numbers.
k -> a,b,c,d...? in its prime factors
If k -> a,b,c,d then k will not be a perfect square
If k -> a²,b²,c²,d² then k will be a perfect square
Insufficient

(3) Combined:
k -> 2²,b,c,d...?
If -> 2²,b,c,d then k will not be a perfect square
If -> 2²,b²,c²,d² then k will be a perfect square

So both statements are insufficient