Tough and Tricky questions: Divisibility/Multiples/Factors.
How many factors does \(x\) have, if \(x\) is a positive integer?
(1) \(x = p^n\), where \(p\) is a prime number
(2) \(n^n = n + n\), where \(n\) is a positive integer
Official Solution:How many factors does \(x\) have, if \(x\) is a positive integer? We cannot easily rephrase the question. Note that we may not need to know \(x\) in order to know how many factors it has.
Statement (1): INSUFFICIENT. Without knowing the value of \(n\), we cannot determine the number of factors \(x\) has.
Statement (2): INSUFFICIENT. This statement by itself is unconnected to the question, because the statement involves only the variable \(n\), whereas the question only involves the variable \(x\).
Statements (1) and (2) TOGETHER: SUFFICIENT. First, we should analyze the second statement further, to see whether we can find a unique value of \(n\).
Since \(n\) is a positive integer, we can test simple positive integers in an organized fashion, checking for equality of the two sides of the equation.
\(1^1 = 1 + 1\)? No.
\(2^2 = 2 + 2\)? Yes.
\(3^3 = 3 + 3\)? No.
\(4^4 = 4 + 4\)? No.
Notice that the left side of the equation is growing at a much faster rate than the right side, so the equation will not be true for any higher possible values of \(n\). Thus, we can determine that the value of \(n\) is 2.
Now, we do not know the value of \(p\), nor of \(x\), but we do now know that \(x = p^2\), with \(p\) as a prime number. Since a prime number has no factors other than 1 and itself, we can see that \(x\) has no factors other than 1, \(p\), and \(p^2\). Thus, \(x\) has exactly 3 factors, and we can answer the question definitively.
Answer: C.
@Bunel, if we know P is prime, and all prime numbers have only three factors, and exponents do not produce new factors, why cant we say that # factors for N = 3 since N itself the just a prime number to a power?