l0rrie wrote:
If x > 1, what is the value of integer x?
(1) There are x unique factors of x.
(2) The sum of x and any prime number larger than x is odd.
First of all, it is not a very straight forward question. Definitely needs some thinking so relax...
Ques: What is the value of x?
So we are looking for a single value of x.
First consider statement 2 since it is easier.
x + prime number greater than x = odd
There will be many many prime numbers greater than x. All prime numbers are odd except 2. So if you can add any prime number greater than x to x and get an odd number, it means x must be even. (because Even + Odd = Odd)
So all statement 2 tells you is that x is even. It could be 2 or 4 or 6 etc
Now look at statement 1.
There are x unique factors of x. Think of the first number greater than 1.
2 has 2 unique factors: 1 and 2 ( 2 is a prime number)
What about 3? It has 2 unique factors: 1 and 3 (a prime number)
4 has 3 unique factors: 1, 2 , 4
Is it possible that any greater number x has x unique factors? No. Why?
For x to have x unique factors, each number from 1 to x must be a factor of x. Say if 10 had 10 unique factors, each number 1, 2, 3, 4, 5..., 9,10 would have to be a factor of 10 (because factors are always positive integers)
But can 9 be a factor of 10 i.e. can (x-1) be a factor of x? No. 2 consecutive positive integers share only one common factor i.e. 1. Why? Check out the post given below for the answer:
question-from-practice-exam-78880.html#p847817So statement 1 is enough to tell us that x is 2.