GMAT Prep Challenge: Can You Do This Math Problem?
No introduction to this onesee what you can figure out and then well talk about it!
This is a GMATPrep problem from the free exams.
* If n and y are positive integers and [pmath]450y = n^3[/pmath], which of the following must be an integer?I. [pmath]y/{3*2^2*5}[/pmath]
II. [pmath]y/{3^2*2*5}[/pmath]
III. [pmath]y/{3*2*5^2}[/pmath]
(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III
In previous weeks, weve been talking about taking a First Glance on Sentence Correction problems. This First Glance works very nicely on quant problems, too!
When the question popped up on the screen, my very first thought was, Ugh, roman numeral!
I hate roman numeral questions and you should, too. They require you to do the work of three problems for the price of one! Whenever I see a roman numeral question, I immediately ask myself, Okay, how hard is the problem? Because if its hard in general, then Im not doing it. Itll take too much time and brain energy.
In other words, my skip! threshold is lower for a roman numeral problem. I have to spot a shortcut or know that the work is not too hard in order to be willing to bother with this kind of problem.
So, my First Glance also tells me that the three roman numeral statements are awfully similar. Maybe I can use that to my advantage. Time to read the problem.
Hmm, Im dealing with integers. I dont understand the significance of that formula yet. Theyre asking what must be an integer
Oh, I get it! The three statements are all fractions and the denominators consist of prime numbers. In order for one of these to be an integer, the numerator, y, has to be divisible by the primes in the denominator. This is a divisibility problem!
Great, I know whats going on and the three statements are similar enough that, once I crack the problem, it shouldnt take me too long to evaluate the statements.
Okay, back to that weird formula. If [pmath]450y = n^3[/pmath], then [pmath]y={n^3}/450[/pmath]. They told me y is an integer, so [pmath]n^3[/pmath] has to be divisible by 450. Time to break that 450 down into its primes.
[pmath]450 = (45)(10) = (3)(3)(5)(2)(5)[/pmath]
And heres where another cool recognition trick comes into play. Ive seen this on other problems before and remembered it. When youre trying to work out divisibility and factors for a squared or cubed number, the factors have to come in pairs (squared) or triplets (cubed).
Go back to the original form of the equation:
[pmath]450y = n^3[/pmath]
[pmath](3)(3)(5)(2)(5)(y) = (n)(n)(n)[/pmath]
Now, each n has to be identical to the other ns, and n itself is an integer, so heres the cool trick: If you know you have any certain prime factor, then you have to have three of those prime factors, one for each n.
In other words, the left side of the equation must include at least three 3's. Only two 3's are included in the 450 figure, so that other 3 must be in y.
Likewise, you must have three 2's and three 5's.
Therefore, y has to contain those extra factors that are missing from 450: 3, 2, 2, and 5.
Great! Well, if y contains 3, 2, 2, and 5, then y must be divisible by those numbers. Check out the roman numerals.
I. [pmath]y/{3*2^2*5}[/pmath]
Yes, y is divisible by 3, 2, 2, and 5. Roman numeral I must be an integer.
Cross off answers (A), (C), and (D).
II. [pmath]y/{3^2*2*5}[/pmath]
Its true that y is divisible by 3, 2, and 5, but y may not contain a second 3. I cant say that this one must be a integer, just that it could be.
Check out the answers again. Answer (E) is also wrong, so (B) must be the correct answereven though you havent evaluated statement III yet!
Now, you have a choice: you can pick (B) and move on, or you can check the third statement, just to make sure you did things correctly. If you have the time, go ahead and check. If youre already over time, pick (B) and move on.
III. [pmath]y/{3*2*5^2}[/pmath]
As with roman numeral II, roman numeral III could represent an integer but doesnt have to. The variable y does have to contain at least one 5, but it may not contain two.
The correct answer is (B).
Key Takeaways: Roman Numeral questions
(1) Be biased. These are often not worth your time. Look for shortcuts or ways to recognize what to do; if the problem looks like its going to be hard and really require you to do three times the work, forget it. Pick your favorite letter and move on.
(2) There are often ways to capitalize on earlier work, though, as we saw in this problem. We had to do a decent amount of upfront work to figure out what to do with the roman numeral statements, but then evaluating those statements was pretty quickand that makes sense because the statements are so very similar.
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.