If X is greater than two, is X the square of an odd prime integer?

Question

If positive integer X is greater than two, is X the square of an odd prime integer?

(1) X has exactly one composite number as factor.
(2) X has odd number of positive factors.

JerryAtDreamScore · Answer

Statement 1 Alone:

A composite number is a product of two positive integers. X could be a square of a prime number or a product of two different prime numbers. Then statement 1 alone is insufficient.

Statement 2 Alone:

This only tells us X is a square. We don't know if X is a square of a prime integer. Then statement 2 alone is insufficient.

Both Statements Combined:

X has to be a square of a prime number. However this square could be 4 or 9, so X doesn't have to be odd. Combined it is insufficient.

Answer: E

hadimadi · Answer

Hi,

(1) Take x=6, it has only 6 as composite factor, but doesn't fulfill requirements
(2) Take x=6, has exactly 3 factors, but doesn't fulfill requirements

(1) and (2): Take 21, has one composite factor with 21 itself, has exactly 3 factors in total with 3,7,21, but it isn't the square of any odd prime integer. Take 49, it has one composite factor with 49, 3 factors with 3,7,49, and it is the square of an odd prime, so we have yes and no as an answer, hence (E)

AJ2606 · Answer

I chose C as the answer and here is my reasoning. 
Given : x > 2 and X is an integer.

(1) X has exactly one composite number as factor.
eg : If x = 6 has factors (1,2,3,6) out of which 6 is the only composite factor but 6 is not the square
If x = 9 has factors (1,3,9) out of which 9 is just the composite factor and it is square. 
Insufficient.

(2) X has odd number of factors.
Any square of an integer will always have odd numbers of factors 
Insufficient.

(1) + (2) 
we know x has odd numbers of factors, thus x is a square and it also has just one composite factor. 
So, x cannot be even at least because an even number >2 is always composite and thus x & X^2 both will be composite
X is odd then 2 possibilities: x is prime or x is composite. 
If x is composite - then x^2 will have at least 2 composite factor (x , x^2) - Possibility rejected.
 (eg : x = 9, X^2 = 81 and factors of x^2 = 1,3,9,27,81) - 3 composite factors 
If x is prime - Let say 3 then x^2 will be 9 (factors : 1,3,9) will always have only 1 composite factor.

Thus (1) + (2) Sufficient.

Bunuel , Please help where my reasoning is wrong.

Rishitha0311 · Answer

Can any of the experts please help with this question? I've opted C as the answer for this.

Thanks in advance!

Bunuel · Answer

If positive integer x is greater than two, is x the square of an odd prime integer?

The question asks whether x = (odd prime)^2.

(1) x has exactly one composite number as factor.

A composite number is a positive integer greater than 1 that is not prime. x having only one composite factor implies that x itself is that composite factor, meaning x = prime^2 or x = (prime_1)(prime_2).

In the first case, its factors are 1, a prime, and X = prime^2, with X = prime^2 being the only composite factor of X. For example, X can be 2^2 = 4. In this case, its factors are 1, 2, and 4, with 4 being its only composite factor.

In the second case, its factors are 1, (prime_1), (prime_2), and x = (prime_1)(prime_2), with x = (prime_1)(prime_2) being the only composite factor of x. For example, x can be 2*3 = 6. In this case, its factors are 1, 2, 3, and 6, with 6 being its only composite factor.

Hence, from this statement, x can be (odd prime)^2, (even prime)^2, or (prime_1)(prime_2). Not sufficient.

(2) x has odd number of positive factors.

Only the squares of integers have an odd number of positive factors, so this statement implies that x = integer^2. However, we don't know whether that integer is an odd prime or not. Not sufficient.

(1)+(2) Since from (2) we got that x = integer^2, then from (1) we are left with the first case: x = prime^2. However, we still don't know whether x = (odd prime)^2 or x = (even prime)^2. For example, consider x = 9 for a YES answer and x = 4 for a NO answer. Not sufficient.

Answer: E.

If X is greater than two, is X the square of an odd prime integer?

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If X is greater than two, is X the square of an odd prime integer?

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