Bunuel wrote:

If x is a positive integer, what is the number of different positive factors of 39x ?

(1) x is a two-digit number

(2) x² has 3 positive factors

------ASIDE---------------

Key concept: If the

prime factorization of N = (p^

a)(q^

b)(r^

c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (

a+1)(

b+1)(

c+1)(etc) positive divisors.

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

---------------------------

Given: x is a positive integer Target question: What is the number of different positive factors of 39x ? Statement 1: x is a two-digit number Let's TEST some values of x:

Case a: x = 11. So, 39x = (3)(13)(11) = (3

¹)(13

¹)(11

¹). So, the number of positive divisors = (

1+1)(

1+1)(

1+1) = 8. The answer to the target question is

39x has 8 positive divisorsCase b: x = 13. So, 39x = (3)(13)(13) = (3

¹)(13

²). So, the number of positive divisors = (

1+1)(

2+1) = 6. The answer to the target question is

39x has 6 positive divisorsSince we cannot answer the

target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x² has 3 positive factorsThis tells us that x must be a

prime numberIMPORTANT: Notice that we can use the same x-values we used to show that statement 1 is not sufficient:

Case a: x = 11. So, 39x = (3)(13)(11) = (3

¹)(13

¹)(11

¹). So, the number of positive divisors = (

1+1)(

1+1)(

1+1) = 8. The answer to the target question is

39x has 8 positive divisorsCase b: x = 13. So, 39x = (3)(13)(13) = (3

¹)(13

²). So, the number of positive divisors = (

1+1)(

2+1) = 6. The answer to the target question is

39x has 6 positive divisorsSince we cannot answer the

target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined IMPORTANT: Since I was able to use the

same counter-examples to show that each statement ALONE is not sufficient, the same counter-examples will satisfy the two statements COMBINED.

Case a: x = 11. So, 39x = (3)(13)(11) = (3

¹)(13

¹)(11

¹). So, the number of positive divisors = (

1+1)(

1+1)(

1+1) = 8. The answer to the target question is

39x has 8 positive divisorsCase b: x = 13. So, 39x = (3)(13)(13) = (3

¹)(13

²). So, the number of positive divisors = (

1+1)(

2+1) = 6. The answer to the target question is

39x has 6 positive divisorsSince we cannot answer the

target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,

Brent

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