How many positive factors does the positive integer x have?

Question

(1)  x  is the product of 3 distinct prime numbers.

(2)  x  and  3^7  have the same number of positive factors.

BrentGMATPrepNow · Answer

Target question : How many positive factors does the positive integer x have? ------------------------------------------ Aside: there's a nice rule for finding the number of positive divisors of a number. If N = (p^ a )(q^ b )(r^ c )..., where p, q, r,...(etc.) are prime numbers, then the total number of positive divisors of N is equal to ( a +1)( b +1)( c +1)... Example : 14000 = (2^ 4 )(5^ 3 )(7^ 1 ) So, the number of positive divisors of 14000 = ( 4 +1)( 3 +1)( 1 +1) = (5)(4)(2) = 40 For more on this concept, see our free video: https://www.gmatprepnow.com/module/gmat- ... /video/828 ----------------------------------- Statement 1 : x is the product of 3 distinct prime numbers. Let a, b and c be the three DISTINCT prime numbers. So, x = (a^ 1 )(b^ 1 )(c^ 1 ) So, the number of positive divisors of x = ( 1 +1)( 1 +1)( 1 +1) = (2)(2)(2) = 8 Since we can answer the target question with certainty, statement 1 is SUFFICIENT Statement 2 : x and 3^7 have the same number of positive factors. Since we COULD apply our rule to find the number of positive factors of 3^7 , we COULD answer the target question with certainty. So statement 2 is SUFFICIENT Answer = D Cheers, Brent

MK1480 · Answer

I found this question interesting as well, and was about to post the same! thanks !

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution

How many positive factors does the positive integer x have?

(1) x is the product of 3 distinct prime numbers.

(2) x and 3^7 have the same number of positive factors.
 
There is 1 variable (x) in the original condition. In order to match the number of variables and the number of equations, we need 1 equation. Since the condition 1) and 2) each has 1 equation, there is high chance that D is the answer.

In case of the condition 1), we can get x=2*3*5. The number of distinct factors is (1+1)(1+1)(1+1)=8. The answer is unique and the condition is sufficient.
In case of the condition 2), if x and 3^7 have the same number of positive factors, then the number of distinct factors is (7+1)=8. The answer is unique and the condition is sufficient. Therefore, the answer is D.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

TheKingInTheNorth · Answer

Hi Bunuel ,

I have doubt in Statement .. How can statement 1 be sufficient .

As per my understanding ...

(1) x is the product of 3 distinct prime numbers.

x is product of 3 distinct prime ...but the power of prime is not known .

Please advise .

Regards

Bunuel · Answer

We cannot say that 2*3^2*5 is a product of 3 distinct primes. The first statement implies that x = prime*prime*prime.

aj12345 · Answer

HI brunel,

I am still having some trouble understanding statement 1:

1) x is the product of 3 distinct prime numbers.

in the example above we used 2x3x5 = 30 (this has 8 positive factors)

what is we use 1x2x3 = 6 ( This has 4 different factors)

wouldn't this make statement 1 insufficient?

Bunuel · Answer

1 is not a prime number.

stonecold · Answer

Nice One
Here statement 1 is sufficient as the number of prime factors = product of powers of primes after raising them by 1
hence sufficient
statement 2 is sufficient as Number of factors =8
hence D

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How many positive factors does the positive integer x have?

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GMAT Data Sufficiency (DS) Questions

How many positive factors does the positive integer x have?

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