The Integer Constraint | Number Properties

We know that the essentials of the GMAT Quant section are pretty simple: advanced topics such as derivatives, complex numbers, matrices and trigonometry are not included, while fundamentals we all learned from our high school math books are included. So it would be natural to think that the GMAT Quant section should not pose much of a problem for most test-takers (especially for engineering students, who have actually covered far more advanced math during their past studies).

Hence, it often comes as a shock when many test-takers, including engineering students, receive a dismal Quant score on the first practice test they take. Of course, with practice, they usually wise up to the treachery of the GMAT, but until then, the Quant section is responsible for many a nightmare!

Today, let’s see what kind of treachery we are talking about – problems like this make some people laugh out loud and others pull at their hair!

Is the product ${pqr}$ divisible by 12?

Statement 1: ${p}$ is a multiple of 3

Statement 2: ${q}$ is a multiple of 4

This seems like an easy {C} (Statements 1 and 2 together are sufficient, but alone are not sufficient), doesn’t it? $p$ is a multiple of 3 and $q$ is a multiple of 4, so together, $p\times q$ would be a multiple of $3\times4 = 12$ . If $p \times q$ is already a multiple of 12, then obviously it would seem that $p \times q \times r$ would be a multiple of 12, too.

But here is the catch – where is it mentioned that $r$ must be an integer? Just because $p$ and $q$ are integers (multiples of 3 and 4 respectively), it does not imply that $r$ must also be an integer.

If $r$ is an integer, then sure, $p \times q \times r$ will be divisible by 12. Imagine, however, that $p = 3, q = 4$ and $r = \frac{1}{12}$ . Now the product $p \times q \times r = 3 \times 4\times (\frac{1}{12}) = 1$ . 1 is not divisible by 12, so in this case, $pqr$ is not divisible by 12. Hence, both statements together are not sufficient to answer the question, and our answer is in fact (E)!

This question is very basic, but it still tricks us because we want to assume that $p$ , $q$ and $r$ are clean integer values.

Along these same lines, let’s try the another one:

If ${10^a \times 3^b \times 5^c = 450^n}$ , what is the value of ${c}$ ?

Statement 1: ${a}$ is 1.

Statement 2: ${b}$ is 2.

The first thing most of us will do here is split 450 into its prime factors:

$450 = 2 \times 3^2 \times 5^2$

$450^n = 2^n \times 3^{2n} \times 5^{2n}$

And do the same thing with the left side of the equation:

$10^a \times 3^b \times 5^c = 2^a \times 3^b \times 5^{a+c}$

Bringing the given equation back, we get:

$2^a \times 3^b \times 5^{a+c} = 2^n \times 3^{2n} \times 5^{2n}$

Statement 1: $a$ is 1.

Equating the power of 2 on both sides, we see that $a = n = 1$ .

$a + c = 2n$ (equating the power of 5 on both sides)

$1 + c = 2$

$c = 1$

Statement 2: $b$ is 2.

Equating the power of 3 on both sides, we see that $b = 2n = 2$ , so $n = 1$ .

If $n = 1$ , $a = 1$ by equating the powers of 2 on both sides.

$a + c = 2n$ (equating the power of 5 on both sides)

$1 + c = 2$

$c = 1$

So it seems that both statements are separately sufficient. But hold on – again, the variables here don’t need to be cleanly fitting integers. The variables could pan out the way discussed in our first problem, or very differently.

Say, $n = 1$ . When Statement 1 gives you that $a = 1$ , you get $10^1 \times 3^b \times 5^c = 450^1$ .

$3^b \times 5^c = 45$

Now note that value of $c$ depends on the value of $b$ , which needn’t be 2.

If $b = 3$ , then $3^3 \times 5^c = 45$ .

$5^c = \frac{45}{27}$

$c$ will take a non-integer value here.

$c = .3174$

The question does not mention that all variables are integers, therefore there are infinite values that $c$ can take depending on the values of $b$ . Similarly, we can see that Statement 2 alone is also not sufficient. Using both statements together, you will get:

$2^a \times 3^b \times 5^{a+c} = 450^n$

$2^1 \times 3^2 \times 5^{1 + c} = 450^n$

$5^{1 + c} = \frac{450^n}{18}$

By now, you’ve probably realized that depending on the value of $n$ , $c$ can take infinite different values. If $n = 1, c = 1$ . If $n = 2, c = 4.8$ . And so on… We don’t need to actually find these values – it is enough to know that different values of $n$ will give different values of $c$ .