I'm having problem understand the unique factors of a number. How is the first statement in the given example sufficient?

Hi gster,
"Unique factors" mean how many different, or distinct, numbers will evenly divide into our original number? For example, 4 has three unique factors: 1, 2, and 4. Even though 2 will divide into 4 twice, we only count 2 once as a unique factor.

In this problem, we know x > 1 and an integer, so x could be 2, 3, 4, and so on . . . . The question is what is x?

1) There are x unique factors of x.

Hmm. What does this mean? It means that whatever x is, it has that number of unique factors. Let's try out some numbers:

2--If x is 2, it has 2 unique factors (1,2). Looks good. The number itself and the number of unique factors match.
3--If x is 3, it has 2 unique factors (1,3). No match.
4--If x is 4, it has 3 unique factors (1, 2, 4). No match.
5--If x is 5, it has 2 unique factors (1,5). No match.
6--If x is 6, it has 4 unique factors (1, 2, 3, 6). No match.

As you go, you will see how hard it is to have as many unique factors as the number x itself. For example, there's no way the number 20 will have 20 unique factors. Because of this, you can see that the only possible value for x is 2.

_________________
Jamie Nelson
ManhattanGMAT Instructor