GMATinsight wrote:
Is Integer x a prime number?
1) x has only one odd factor
2) x has only one even factor
Source:
https://www.GMATinsight.comI like this problem! It's a little unusual - I haven't seen many GMAT problems that talk about 'odd factors' and 'even factors'. However, you can work it out using only basic math and what you know about the definition of factors. I'd say you could definitely see something like this on the GMAT.
I'd probably start by 'translating' the question and the statements. That's a good place to start with number properties problems (problems that ask about issues like primes, factors, divisibility, etc.) You really want to focus in on what each piece of the problem is trying to say. When the question asks 'is x a prime number?' it's asking something like "is x 2, 3, 5, 7, 11, 13, ...?, or is it a different number instead?" Briefly remind yourself that 1 isn't prime - that's often relevant in these problems!
Statement 1 says that x only has one odd factor. What kinds of numbers only have one odd factor? Well, 1 is an odd number, and it's a factor of everything. So, every number has at least one odd factor! The trick is to find a number that has
only one odd factor. That means it doesn't have any
other odd factors, besides 1.
This should make you think about 2, because it's a special number - it's the only even prime. x could equal 2, since it doesn't have any odd factors other than 1. That would mean the answer to the question was 'yes'.
What other numbers have no odd factors (besides 1)? Okay, that would have to be a number that's only divisible by 2, 4, 8, etc. (It couldn't be divisible by 6, because then it would be divisible by 3! Likewise, it couldn't be divisible by 10, because then it would be divisible by 5). The simplest number to look at first is 4, which fits. It also isn't prime, so the answer is 'no'.
We got a 'yes' and a 'no', so 1 is insufficient.
Statement 2 is trickier. We know that 1 has to be a factor, since it's a factor of every number. We also know that 2 has to be a factor. Think about it this way: if a number has an even factor, it's definitely divisible by 2. So 2 must be the
only even factor of the number. What kinds of numbers fit that?
The number 2 works, and it's prime.
Bigger numbers, like 4 or 6, don't work. The problem is, every number is a factor of itself. So, 4 has two different even factors, 2 and 4. 6 does as well (2 and 6). That means no even number other than 2 will fit the statement. (2) is sufficient.