M30-01

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

Official 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 . 
 
Note that all odd primes satisfy this statement, as does the integer 4 (4 does not have a factor p such that  2 < p < 4 ). This information alone is not sufficient to determine whether  x  is a prime number.
 
(2) The product of any two factors of  x  is greater than 2 but less than 10. 
 
This implies that  x  could be 3, 5, or 7. This condition alone is sufficient to determine that  x  must be a prime number.
 
 Observe 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, which is not greater than 2 as required by the statement. Also, note that  x  cannot be 9 because 3 and 9 are both factors of 9, and their product, 3*9=27, is greater than 10.

Answer: B

zet · Answer

Could you explain the S2 in detail? I could not understand.

If product of any 2 factors is less than 10; then they can be 1*9 which means x is not a prime.

chetan2u · Answer

Hi 
Statement II talks of any two factors, but you are just looking at two factors here ...
So if you are looking at 9..
Factors are 1,3,9..
So if you pick 3 and 9, the product is 3*9=27..
1*3 and 1*9 may be less than 10 but we are looking at any two
Hope it helps

kupratik1 · Answer

if x = 6, then product of its factors will be 2 x 3 = 6 which satisfies the statement but x is not prime. kindly clarify

amanvermagmat · Answer

Hi

The Statement 2 states that product of  any two factors of x  is greater than 2 but less than 10. This means that no matter which two factors of x you take, their product should not exceed or be equal to 10. Also, no matter which two factors of x you take, their product should not be less than or equal to 2.

So x=6 will NOT satisfy this property. because I can state two such factors of 6 (3 and 6) which multiply to give product greater than 10.

Similarly, x=4 will also NOT satisfy this property, because I can state two factors of 4 (1 and 2) which multiply to give a product equal to 2.

The required number x should be such that we should not be able to come up with any two factors of x, which multiply to give 2 or less.... NOR should we be able to come up with any two factors of x which multiply to give 10 or more.

Manoraaju · Answer

Hello Sir,

I have great difficulty in understanding S1. Could you please provide a detailed explanation?

Thank you.

Bunuel · Answer

(1) states that when x is factorized, it won't have a factor that is greater than 2 but less than x itself. By definition, a prime number is a positive integer with exactly two factors: 1 and itself. Consequently, any odd prime will satisfy this condition. However, not only odd primes fulfill this requirement; the integer 4 also does not have a factor greater than 2 but less than 4 itself (since the factors of 4 are 1, 2, and 4). As a result, both odd primes and the integer 4 satisfy (1), rendering this statement insufficient to determine if x is a prime number.

Hope it's clear.

alainca · Answer

This is a good question and the explanation is clear. Thank you!

BobsterGMAT · Answer

I think this is a high quality question and it has an excellent answer! kudos!

Sriparnacj · Answer

I think this is a  high-quality  question and I agree with explanation.

coreyander · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Mayank2707 · Answer

If x=4 is put in s2, this satisfies the condition.Can someone explain why x=4 is not the solution considering s2 alone?

Bunuel · Answer

This is explained in detail in the highlighted part above.

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

For x=4, the factors are 1, 2, and 4. The product of 1 and 2 is 2, which isn't greater than 2. Thus, 4 doesn't satisfy the second statement.

M30-01

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