GMAT Prep - Even/Odd

rosh26 wrote:If each of the prime factors of x^2 is odd, why can't there be other even integers that are just its prime factors, not actual factors?

rosh26 wrote:got it! I didnt realize every factor of x^2 is odd if all the prime factors are odd...

thanks!

beeparoo wrote:Rosh,

In order to get an even value, you need either 1st term of expression, or, the 2nd term (or even both) to be even.

So, either (x^2 + 1) is even, or (x + 5) is even

Statement 1: x is odd, therefore x^2 is odd. Odd + 1 = Even. (Likewise, odd + 5 is also even)
SUFF

Statement 2: If every prime factor of x is greater than 7, then each factor is odd. As a result, x is odd.
SUFF

Choose D

I hope it's clear!

Sandra

ildude02 wrote:Does this mean, x^2 can only be prime number ?
No, I don't know what led you to think this. If x alone is an integer, there is no way that x*x is a prime number; it completely contrasts the definition of a "Prime Number".

Also, in general, is there anything that stands out with regards to the factors of x^2 when compared to the factors of x? Like can we assume, if every factor of x^2 will also be a factor of x or not?
Every factor of x*x is not necessarily a factor of x. For example: Let x = 4. Then x*x = 16, which has 1, 2, 4, 8, 16 as factors. While 1, 2, and 4 are factors of x = 4, the values 8 and 16 are not factors of 4.

x and x*x will have common prime factors, BUT the quantity of primes will differ as you can see from the example above.

Sometimes, I find the arithemetic involving factors a little tricky and I was hoping if there are any set of rules that would help me with it. Appreciate your response
What types of rules in particular do you seek? Numbers and their factors come with all sorts of different manipulations that make for good GMAT brain-teasers. I highly recommend doing as many problems as possible to get a feel for questions that pertain to prime numbers. Those are the best ones for developing good factoring/multiplication skills.

ildude02 wrote:Hi beeparoo,

With regards to x^2 being a prime number, I meant that in context with statement 2. Statement 2 says, x^2 has prime number factor greater than 7. So I beleive, x^2 should be an integer greater than 49, correct?
Yes. x^2 can be 43681 = (11*19)^2, for instance.

One part of the statement 2 that I'm not sure was, does it mean, x^2 ONLY has prime number factor GREATER than 7?
Yep

That would mean, it should not have any prime fatcors like 2, 3 5 or 7.
Correct! But it can have factors like 11, 13, 17, 19, etc.

That implies, it would leave x^2 with only prime numbers greater then 49 such as (53^2, 59^2 , etc)which has only 3 factors, 1, sqrt of x^2 and x^2, where sqrt of x and x^2 are both prime numbers greater than 7.
Noooooooooo. You are effectively equating "x = a factor of x". Suppose that x = 143 = 11*13. Then x^2 = 11*11*13*13. Here, x^2 has 6 different factors, 4 prime factors altogether, and 2 distinct prime factors.

If you want to pretend that x^2 = 53^2 or 59^2, it is OK to do so. But I can tell where your mind is creeping and I think you are discounting some things.

Hence, I can only see x^2 being a prime number, when combined with statement 2. Is my reasoning correct or did I miss something?
No! x^2 is not a prime number; it is comprised of prime number factors. Further, it has twice the amount of prime factors as x.

If every prime factor of x, or x^2, is greater than 7, then you can determine that x is odd.
(Because all primes other than 2 is an odd integer).