Numbers Property Strat. Guide - Chp. 2 - Odds,Evens,Pos,Negs

[clarence.booth]

Hi,

My question refers to pg. 35 in the Num. Prop. Strategy Guide for the following Data Sufficiency example:

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.

My problem involves only statement 1, both just understanding what it means and how it is sufficient. The explanation provided in the text is going over my head. Here it is for your reference:

"Statement 1 tells you that there are x unique factors of x. In order for this to be true, every integer betw. 1 and x, inclusive, must be a factor of x. Testing numbers, you can see that this property holds for 1 and for 2, but not for 3 or for 4. In fact, this property does not hold for any higher integer, because no integer x above 2 is divisible by x-1. Therefore, x=1 or 2. However, the original problem stem told you that x>1, so x must be equal to 2. SUFFICIENT."

Thanks.

[RonPurewal]

"Factors" are positive integers.

If x has x factors, that means that ALL integers from 1 through x must divide evenly into x. (E.g., if you wanted 11 to have 11 factors"”"”which it doesn't, of course"”"”then you'd need every single one of 1, 2, 3, ..., 11 to go evenly into it.)

If you investigate a little, you'll quickly figure out that only x = 1 and x = 2 satisfy this condition. (x = 1 is excluded by the original restrictions on the problem.)

What they're doing here is pointing out that, for any value of x greater than 2, the next lower integer can't be a factor. E.g., 2 is not a factor of 3; 3 is not a factor of 4; and so on.
If it's hard to understand, it's only hard because there are so many letters flying around that it starts to look like nonsense.

[clarence.booth]

Thx, Ron, this makes sense. I did have to read (and re-read) your response as working with the variable x made things confusing, but I get your logic.

I'm curious: when I first came across this problem I immediately thought to myself how I could make solving much easier if I tested numbers, but I simply didn't know which number/s were best to test...5? 7? The example started with 1 and 2. How would I have known intuitively to test 1 or 2 as oppose to any of random number/s? What would have tipped me off in the problem that 1 and 2 would have been a good place to start testing?

[tim]

*One* general strategy for testing numbers is to test extremes, and this problem would be a good one for this strategy. Start small and then work your way up. You'll realize almost immediately that nothing larger than 2 can work.

[RonPurewal]

Thx, Ron, this makes sense. I did have to read (and re-read) your response as working with the variable x made things confusing, but I get your logic.

I'm curious: when I first came across this problem I immediately thought to myself how I could make solving much easier if I tested numbers, but I simply didn't know which number/s were best to test...5? 7? The example started with 1 and 2. How would I have known intuitively to test 1 or 2 as oppose to any of random number/s? What would have tipped me off in the problem that 1 and 2 would have been a good place to start testing?

Wherever it's possible to impose organization, impose organization. Take chaos, and put it into order.

In this case, we're restricted to positive integers.
Positive integers have a natural organization"”"”the one you learned in pre-school: one, two, three, etc.
So, just use the natural order!

1 is off-limits, so just try 2, 3, 4, ...

The GMAT won't hide patterns from you, by the way. If you are trying cases in an order that's reasonably organized, you'll find what you need to find very quickly.

[clarence.booth]

Got it, very helpful. Thanks, Ron.

[tim]

Glad to hear it!