Really cool question, and great explanations!
One of my favorite trends on the GMAT is its use of more 'unique' number properties like that in statement 2. It's not an even/odd, positive/negative, or units digit "known" property, but you can try small numbers to arrive at a pattern and establish that statement 2 is only true for odd numbers. I like to look at "Find the Pattern" number properties in two parts:
1) Use small numbers to see if you can find a pattern.
Here, as Shankar did, you can try a few sequences of odds and evens to see if there's a pattern:
Odds: 1 + 2 + 3 --> sum of 6, which is divisible by 3
1 ---> the sum of just one integer is divisible by 1 (easy, but ok)
1 + 2 + 3 + 4 +5 ---> sum of 15, which is divisible by 5
It seems like the rule works for odd numbers. But evens?
1 + 2 --> 3, which is not divisible by 2
1 + 2 + 3 + 4 --> 10, which is not divisible by 4
maybe we don't start with 1...how about: 2 + 3 --> 5, which is not divisible by 2 (and we were adding two integers). It doesn't look like it works for evens, so I think you can be fairly confident here that x must be odd, and pick D.
2) Try to find why the pattern holds (useful more in practice than on the time-sensitive test)
This can actually be a lot of fun, and doing this in practice helps you to really gain an understanding of how numbers and calculations really work. If we try this for an odd number like 5, we have:
1 + 2 + 3 + 4 + 5
Now, note that 5 is going to definitely be divisible by 5, and so we need the other numbers to add to a multiple of 5 in order for the sum to be divisible by 5. Then notice that we can pair the numbers (1+4) and (2+3) to form 5s.
The same thing will work for 7:
1 + 2 + 3 + 4 + 5 + 6 + 7
7 is a multiple of 7, and so are (1+6), (2+5) and (3+4).
And even if we shift the set, we'll still have a rule that holds:
2 + 3 + 4 + 5 + 6 + 7 + 8
(2+5), (3+4), and (6+8) are all multiples of 7
But if we try with evens, we don't have those pairs to use:
1 + 2 + 3 + 4
4 is a multiple of 4, and so is (1+3), but the 2 stands on its own and we can't pair it with another number to get a multiple of 4.
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Now, this may not be the clearest explanation of all time, but what I'm more interested in is just showing the process of trying to establish a reason for a rule. I hadn't really ever though of this rule until seeing this question, but within a couple minutes just doing those quick tests with small numbers I feel like I have a really strong handle on it. And if you make that a part of your study regimen - using small numbers to test/prove number properties, and then trying to analyze why they're true - you can teach yourself more than just one rule, but in addition a style of thinking that will allow you to figure out several similar questions on test day.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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