I’m here with a number theory data sufficiency problem. Like many of the other problems, we’re going to look at this problem over here, structurally, as well as mathematically. Taking a look at the stem the first thing we are struck by is the idea that we need to figure out this evenness and oddness.
1. What Do We Need?
When we ask ourselves what do we need: a few things should draw our attention: First, that one of these elements is squared. So if B is squared then no matter what it is its square will have the same identity: even squared is even, odd squared is odd. But also because we’re adding these two things together, for something to be odd one of them has to be odd, the other has to be even. But it could go either way so there’s a lot of moving pieces. The easiest thing to do is to say: “I need to know if each of them is even or odd.” But, of course, we know that the GMAT is not going to give us this information.
2. Start With Statement 1
Let’s take a look then at statement one and statement two. And because statement one is very straightforward we should begin there. So here we’re told very quickly succinctly: one is even one is odd. We can run a scenario and plug in some numbers, a two and a three for example, or deal with it at the identity level Either way, that gives us a straight answer that is sufficient. If that’s not visible to you I would suggest that you review your number properties. In general, this is enough and we say: “Okay, well, one’s sufficient.”
3. Statement Number 2
Then we get into number two and we have this “B plus C” is odd and immediately we might end up dismissing this and this would be a mistake. The reason we end up dismissing it is because one was so straightforward in addressing what we needed that two feels like because B and C aren’t extricated from each other that it’s almost too complex. So the GMAT may have lulled us into a sense of security with statement number one, which I think is one of the really neat structural features of this problem. If statement one were more complex we would actually spend more time looking at statement two.
Diving into statement two a little more deeply we can see that because B plus C is odd rather than even one must be even the other must be odd. And because it doesn’t matter which, something that we ascertained when we were looking at the question stem which is why that proactive thinking is really important, we can say well as long as one is each then that’s going to be sufficient as well. And so here the answer is D.
4. Further Information
I want to put up a third piece of information. And this is a really useful thing to do when you’re self-prepping is to look at data sufficiency and then postulate what other piece of information might have some subtlety, might the GMAT give us to induce us to an incorrect answer by modulating the complexity not in the question stem but in the introduced information. So here we have C equals B over 2. What this means is B must be even. Take a minute to think about that. We can’t know anything about C but B must be even because they’re integers and because you can slice B into two B is the even one. It’s tempting to move that 2 over and say 2C equals B and say: “Wait, C is even.”
But if you think about that a little more deeply it doesn’t add up because what we’re doing is multiplying C. An odd or an even number times 2 is going to result in an even. So this is a really great problem form because the same pattern of even/odd identities with different embedded equations and different ways of hiding whether B or C or M, N, or X and Y, or P and Q are odd or even is a very common trope especially as you get to the more challenging levels of the GMAT where you have these abstract DS questions, abstract inequalities that are really the bread and butter of 700 plus.
5. Examine This Problem Form Deeply
So as a more general problem form this is one to examine deeply and play around with in a whole bunch of different ways. You can introduce exponents, absolute value, inequalities as I mentioned, quadratic identities are a big one where, for example, you have a difference of two squares and then you’ve got one piece or the other, the X plus Y or X minus Y and they give you information on that. And so as you’re doing this, one of the most important pivot questions to look at is: “How do I convert this piece of information into the information they’re asking me about in the question stem?” or vice versa: “How do I relate this information in the question stem to this piece of information?” because almost certainly they’re going to be related and it’s in that relationship that you determine whether or not it’s sufficient.
And typically as the subtleties increase that relationship is what defines the entire problem. I realize that’s a little meta but these questions are a little meta. So I hope this helps! Wishing you guys a great day and like and subscribe below and we’ll see you real soon!