If x is a positive integer greater than 1, is x a prime numb

Question

If x is a positive integer greater than 1, is x a prime number?

(1) x does not have a factor p such that 2<p<x.

(2) The product of any two factors of x is greater than 2 but less than 10.

Bunuel · Accepted Answer

SOLUTION

If x is a positive integer greater than 1, is x a prime number?

(1) x does not have a factor p such that 2<p<x. Notice that all odd primes satisfy this statement as well as integer 4 (4 does not have a factor p such that 2<p<4). Not sufficient.

(2) The product of any two factors of x is greater than 2 but less than 10. This implies that x can be 3, 5, or 7. Sufficient.

Notice that x cannot be an even number because any even number has 1 and 2 as its factors and the product of these factors is 2, not greater than 2 as given in the statement. Also notice that x cannot be 9 because 3 and 9 both are factors of 9 and 3*9=27>10.

Answer: B.

Donnie84 · Answer

St1: x does not have a factor p such that 2

magneticlp · Answer

Statement 1. What if x = 4 (not prime)? It has factors: 1, 2 and 4. All these factors satisfy inequality 2<p<x. INSUFFICIENT.

Statement 2. If x is 9, then 3*9 = 27 but the statement says that the product of ANY two factors of x is less than 10. So, there are only 3, 5 and 7 that satisfy to restriction. SUFFICIENT.

IMO correct answer is B.

Donnie84 · Answer

Thanks magneticlp. Your solution is indeed correct. I need to be more careful :cry:

manpreetsingh86 · Answer

Statement 1: x does not have a factor p such that 2<p<x.

lets say x=10, then x has 1,2,5,10 as its factors but as per the statement 1, x does not have a factor p such that 2<p<x. hence x cannot be 10. 
Let say x=7 then x has 1 and 7 as its factors and it completely satisfy the condition 1, hence x can assume 7 as a value.

Moreover we know that prime nos. are divisible by 1 and the number itself, hence only numbers which are going to satisfy the condition mentioned in statement 1 will be prime nos. hence statement 1 sufficient.

Statement 2: The product of any two factors of x is greater than 2 but less than 10
lets say x= 4, then x has 1,2,4 as factors. Now the product of 2 and 4 is greater than 2 and less than 10 hence satisfied the condition. but the product of 1 and 2 doesn't hence x cannot be equal to 4.
similar result holds true for all the non-prime numbers hence x cannot be a non-prime number.

lets say x=7 then x has 1,7 as its factors. product of 1 and 7 is greater than 2 and less than 7 hence holds true. Also , the same result holds true for the prime numbers greater than 2 but less than 10. hence x will be a prime number greater than 2 and less than 10.
Therefore 2 alone is also sufficient to answer the question.

Hence i will go with option D. Each alone is sufficient to answer the question.

jn30 · Answer

1) x = 4 works and so does x = 5. One is prime, one is not. Thus, Ins
2) x = 4 (product of 2*4 = 8) and x = 5 (5*1 = 5). Ins.
(1+2) Can use the same numbers as above. Ins.

(E) is answer.

GMarc · Answer

1) 4 and 11 both fulfill the criteria and one is prime the other is not. So 1) alone is insufficient to answer the question 
2) 1 and x itself are both factors of x so we know that 2 < x < 10. Can a non prime numbers fulfill the condition ? You just have to try all the non-prime (4,6,8,9), for 4,6 and 8 the factor 2 multiplied by 1 is not greater than 2 and for 9 the factor 3 multiplied by 9 is greater than 10. Conclusion => x has to be prime and 2 is sufficient.

Answer is B

sgangs · Answer

The correct anser IMO is E

From Statement 1: If x is prime it is true that the factors of are p, such that 2<p<x . But if x=4 (the only case with non-prime), then also the statement holds good.

From Statement 2: Product of any to factors of x are such that 2<product<10. Now this means the number itself has to be less than 10 (or else x*1=x>10, statement doesn't hold). Similarly x*(any other factor, p)<10. That would reduce our choice to nos<=5. But again, we cannot say whether it'll be prime or not (3,4,5 all falls in this category)

I hope its clear.

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sgangs · Answer

Thanks a lot Bunnel. I had by mistake taken the conditions as >=2 instead of >2 for statement 2 it seems. Thanks a lot again for posting the correct answer

anubhavdogra298 · Answer

hi bunnel,

In statement 2-: why the case of 4 not considered as the factor of 4 are 4,2,1 & product 4&2 is 8 which is greater than 2 but less than 10 so therefore 4 is also satisying the condition along with 3 , 5, 7 finally we can conclude that staement 2 is insuficent.

Therefore the answer should be B

Bunuel · Answer

The product of ANY two factors of 4 is NOT greater than 2, as given in the second statement: 1*2=2. Hence x cannot be 4.

Hope it's clear.

PerfectScores · Answer

Statement I is insufficient:
x = 4 - the factors are 1, 2 and 4. So there is no factor which is between 2 and 4. Hence its a NO.
x = 3 - the factors are 1 and 2. So there is no factor which is between 2 and 4. Hence its a YES (x is a prime number)

Statement II is sufficient:
Since 1 is a factor of all the numbers hence 2 cannot be a factor of that number. Hence the numbers will have to be either 3, 5 or 7.

So Answer is B

anubhavdogra298 · Answer

Hi bunnel 
Still not clear why are you not considering the case 4*2 =8 where 2. & 4 both are factor of 4

Bunuel · Answer

Are you talking about the second statement? If yes, then x cannot be 8 for the second statement for the same reason it cannot be 4: the product of ANY two factors of 8 is NOT greater than 2. The factors of 8 are: 1, 2, 4, and 8. The product of 1 and 2 is 2, which is not greater than 2.

AKG1593 · Answer

Good question.Got confused in the second statement's use of ANY 2 factors.Couldn't see that even nos. are excluded from this statement very subtly.

Posted from my mobile device

hellosanthosh2k2 · Answer

Answer should be B

Statement 1:
no factors of x, such that 2 < p < x
=> no prime numbers will have factors such that 2<p<x
=> Also if x = 4, then also no factors between 2<p<4

Not Sufficient

Statement 2: The product of any two factors of x is greater than 2 but less than 10.

Note: For any even number, we ll have (1,2 ) as factors, product of these factors is 2, but given product of any two factors of x is > 2
=> x is not even number.

now we are left with {3,5,7,9} => Note, we have cannot have odd numbers greater than 9, if were, then prod of two factors will be greater than 10.

if x were 9, prod of factor (3,9) > 10, so 9 is Out.
we are left with {3,5,7} => Prime numbers => Sufficient

zlishz · Answer

Bunuel 
Isn't the second statement become insufficient when we consider 4? Factors of 4 are 1, 2 and 4, any product of 2 factors of 4 will fall between 2 and 10, right?
Also why are we not considering other prime numbers greater than 10?

Thank you!

Bunuel · Answer

4 does not satisfy the second statement, which says that the product of any two factors of 4 is not greater than 2. 1 and 2 are both factors of 4: 1 * 2 = 2, not greater than 2.

If x is a positive integer greater than 1, is x a prime numb

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