Number properties questions are more common on the GMAT Quantitative section than most test-takers would like. In all the years I’ve spent teaching and tutoring this test, I’ve seen few questions approached with more apprehension and dread than questions concerning number properties. This is unfortunate principally because these questions are not at all difficult when addressed patiently and systematically.
It is furthermore the case, however, that the immediate trepidation inspired by a number properties question does little but serve as a distraction and impetus to beat a hasty retreat (or else dawdle unproductively) – which means you’re less likely to get the question right. In general, because students have a hard time separating their perception of their performance on one question from that of other questions, getting discouraged and overwhelmed by a number properties question quite often translates negatively to subsequent questions – which means you’re less likely to get those right, too. What can be done? As indicated above, with respect to questions involving number properties, your method is everything. Laying out a fungible method and consistently sticking to it will invariably lead you through the morass of abstraction to the right answer, no matter how complex the question.
Number properties questions come in several (apparently equally vexing) varieties: there are those concerning odds and evens, those involving positives and negatives, questions regarding primes, as well as any combination of the above! Additionally, these questions can take the form of either a problem-solving question or a data sufficiency question, which further lends itself to exasperation. No wonder so many test-takers are left frustrated or flummoxed.
Before we take a look at any specific questions, let’s lay out a few guidelines. First, it’s essential that the question stem be analyzed and understood by you as much as possible in advance of considering the answer choices, or as is the case with data sufficiency questions, the statements. This means you must avoid the cardinal error of rushing through the question to read either the answer choices or statements. Evaluating the question stem and extrapolating any relevant information first is crucial to your success on these questions.
Secondly, define your approach. Many students tackle number properties questions with a strictly “guess-and-check” strategy that involves selecting numbers to substitute for any variables involved and then checking the truth value of the results. “Picking numbers,” as this strategy is sometimes known, is highly inefficient if not paired with some knowledge of the properties of numbers themselves. This sounds more intimidating than it is. For the purposes of the GMAT, the rules governing the behavior of odds and evens, or positives or negatives, are simply understood; this means that they can be easily memorized or quickly puzzled through on the spot. For example, know in advance that any question regarding prime numbers will likely be invested in determining whether or not you understand that the number 2 is the first prime and the only even prime number. Forgetting to consider this exception to the general rule that primes are odd will almost invariably affect the outcome of questions about primes.
Another thing to contemplate in advance of the test is what makes a number even or odd. Simply put, an even number is a number whose component parts can be wholly accounted for in pairs. All even numbers are simply the concatenation of a series of pairs. The number 8 is four sets of 2; 10 is five sets, etc. Thus all even numbers are universally divisible by 2. Odd numbers, on the other hand, have a dangling “one” or extra number tacked on to the series of pairs that comprises it. For instance, the number 3 has a one tacked on to one pair. Taken together, we can generalize the behavior of odds and evens and understand that:
For addition: O + O = E, E + E = E, and O + E = O.
For subtraction: O – O = E, E – E = E, E – O = O and O – E = O.
Moreover, for multiplication: the only way to produce an odd is with a closed set of odds. As soon as you throw an even into the mix, the result becomes even. O x O = O, O x E = E, and E x E = E. Similarly, it follows that any even number raised to a positive exponent will produce an even number, and any odd number raised to the power of any positive exponent will yield an odd result. Having thought about and understood these rules in advance of the test will ensure that you move through the question as effectively as possible, picking numbers only where necessary to confirm what you already suspect.
Many number properties questions will deal with positives and negatives, taking a concept understood by most test-takers since childhood and complicating it with basic algebra. Again, the key here is understanding how the numbers work in conjunction with one another and being patient and methodical. Consider the following example:
If x, y, and z are integers, and xyz < 0, is y < 0?
1. z^3 < 0
2. x > z
First, analyze the question stem: “integers” means I’m dealing with positive and negative numbers that are not decimal or fractional in nature, so I can rule out fractions (which one must typically keep in mind as examples of numbers that become smaller when raised to a positive exponent). “< 0” is code for negative, so in other words, x times y times z is negative, is y negative? At this point most students would jump to the statements and begin muddling through. The best way to tackle the question, however, is to continue to evaluate the question stem, extracting any and all information from it before I consider the implications of the statements. Namely, xyz < 0 indicates to me that there are four possible scenarios: 1) ++-, 2) +-+, 3) -++, or 4) —. Now I know exactly what combinations of positive and negative numbers are possible, and I can look to the statements specifically for a clarification of those possibilities.
Statement 1 tells me that z^3 < 0 which boils down to z is negative, since a positive number raised to an odd exponent will yield a positive in turn. Where does knowing that z is negative get us? Unfortunately, not as far as we would like, since there are two distinct possibilities laid out above in which z is negative, namely the first and the last. In the first possibility z being negative means y is positive, while conversely in the last scenario, y is also negative. Thus Statement 1 is insufficient, and the answer must be B, C, or E. We must now examine the second statement independently from the first and on its own terms before considering the two statements in combination, if necessary. Statement 2 appears to provide even less information than the first, not even allowing me to determine whether or not x is positive or negative. (Selecting numbers to illustrate that this is the case is simple enough.) Finally, in combination, we know from Statement 1 that z is negative, and Statement 2 tells us that x > z, but once again as picking numbers would indicate, the fact that x is greater than z does not guarantee that x is positive, since x could simply be a “less negative” number than z (like -3 and -4 for x and z, respectively). Hence, the answer must be E, neither statement answers the question of whether y is negative either independently or together.
In sum, PATIENCE and working methodically are key. Study a few rules and moves through these questions with the confidence of the test-taker who knows that preparation makes even the toughest questions manageable!