For nonzero integers M and N, is M/N > (MN)^4?

Karishma, it's from the Veritas Essentials Course handbook. Page 53.

Thanks Yalephd. I will bring it to the notice of the concerned person.

I have some feedback on this topic. The question is actually a slightly modified version of an OG question (the actual OG question is essentially the same as this one). It does not need either statement so yes, technically, there is a question mark. But, it was added in the Essentials course material to provide a great takeaway - something our course excels in. Brian is going to chime in soon on this thread to explain it.

Hey guys,

Great discussion here - and, honestly, that's exactly what the question was designed to do. I'll chime in since I wrote the question and that part of the lesson. The question itself is a pretty straight ripoff of a question from one of the OG books (I think it's the 11th edition...I don't have all my books in front of me though to check). More importantly, the lesson is this:

The way the question is written, the correct answer is D, as either statement (or, as you've found, really any statement) is sufficient to answer the question. But if you simply flip the sign in the question to change the question from:

...is M/N > (MN)^4

to

...is M/N < (MN)^4

the answer goes all the way from D to E. The reason? In the second option, both M and N could be either 1 and 1 or -1 and -1, and that would create a "tie" in which M/N is NOT LESS THAN (MN)^4 - they'd both be 1. The way the question is initially written, the tie is still NOT GREATER THAN, so it provides the same answer "no". But in the second case, the "dominant" answer is "yes" (for all values other than both 1 or both -1 M/N is less than (MN)^4), but the exception case of -1 and -1 - which is allowable by both statements - provides the answer "no". In this case, because flipping the sign takes the "tie" from the same answer to a different one, we can make one small tweak to the question to get the farthest-away possible answer.

So what we're trying to do in that lesson is encourage students to look for small tweaks in DS questions that would then elicit different answers. The more you do this - the more you train yourself to think like the testmaker - the more likely you are to be aware of the subtleties in those questions on test day. Many DS questions are missed simply because people fail to see those details - they see what they want to see or what they expect to see, and aren't fully dialed in to how the GMAT is carefully designed to punish you for those assumptions. The key is awareness - and if you've been training yourself to see the multiple little nuances in how a question can be asked, you'll be aware when you see questions of the variations that each could pose.


So that's the explanation of that question, and if you have that book you'll see that the title on that page and the discussion question immediately after the question are all geared toward teaching that lesson and encouraging that thought process. And I'm happy to see that we got a really healthy discussion going on this thread about that question - as the author of that lesson, I'd say mission accomplished!

Thanks for your input Brian. I am sure it has cleared up all confusion.

Brian / Karishma

Isnt this question similar to https://gmatclub.com/forum/x-is-a-posit ... 87380.html where the stem is sufficient to answer the question. When I saw this question I was wondering whether I should evaluate the statements. And both the statements provide the exact same information (x is integer). Thoughts??

Here, the question stem is not sufficient to answer the question. The question is: Is y divisible by 5?

A positive integral power of 9 has a unit's digit of 9 or 1 depending on whether the power is odd or even (respectively). Since divisibility by 5 is involved, we know that the last digit is enough to tell us whether the sum will be divisible by 5. (If you are not very clear about this, check remainder-concept-111770.html#p904570)
The powers of 9 are consecutive so the last digits will look something like this:
...9 + ...1 + ....9 + ....1 + ....9 + ....1
or
...1 + ....9 + ....1 + ....9 + ....1 + ...9

Either ways, last digit of the sum of the 6 terms above will be 0 so this is divisible by 5.
Now the question is, do we know whether the powers are positive and integral? The question stem tells us that x is positive. So (x+1), (x+2)... all have to be positive. Now, do we know whether the powers are integral? Question stem doesn't tell us this but the statements do.
Both statements individually tell us that the powers are integers so answer would be (D).
If the power is not an integer, all our reasoning goes for a toss. e.g.
\(9^{\frac{1}{4}}\) gives us an irrational number.

Check out the discussion of the answer here: for-nonzero-integers-m-and-n-is-m-n-mn-112949.html#p915063

Can't we have M as -1 and N as -2 (thus satisfying both the statements) in which case M/N > (MN)^4?

After reading the question stem what I inferred was that Is M/N a +ve decimal value. Because in this situation M/N > (MN)^4

So I landed up with (E)

Bunuel VeritasKarishma please could you comment?

When M = -1 and N = -2, we get
M/N = 1/2
and
MN = 1*2 = 2 so (MN)^4 = 2^4 = 16 (note that M and N are multiplied here, not divided)
M/N is not greater than (MN)^4

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

Remember the relation between the Variable Approach, and Common Mistake Types 3 and 4 (A and B)[Watch lessons on our website to master these approaches and tips]

Step 1: Apply Variable Approach(VA)

Step II: After applying VA, if C is the answer, check whether the question is the key question.

StepIII: If the question is not a key question, choose C as the probable answer, but if the question is a key question, apply CMT 3 and 4 (A or B).

Step IV: If CMT3 or 4 (A or B) is applied, choose either A, B, or D.

Let's apply CMT (2), which says there should be only one answer for the condition to be sufficient. Also, this is an integer question and, therefore, we will have to apply CMT 3 and 4 (A or B).

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find the value Is \(\frac{M}{N} > {MN}^4\) - where 'm' and 'n' are non-zero integers

Second and the third step of Variable Approach: From the original condition, we have 2 variables (m and n). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.

But we know that this is a key question [Integer question] and if we get an easy C as an answer, we will choose A, B, or D.

Let’s take a look at each condition.

Condition(1) tells us that \(\frac{M}{N} > 0\).

=> When M/N > 0 then either M and N both are positive or both are negative.

=> For M = 1 and N = 1 => \(\frac{M}{N}\) = 1/1 = 1 and \({MN}^4\) = 1^4 = 1 - \(\frac{M}{N} > {MN}^4\)- NO

=> For M = 4 and N = 2 => \(\frac{M}{N}\) = 4/2 = 2 and \({MN}^4\) = 8^4 - \(\frac{M}{N} > {MN}^4\)- NO


Since the answer is a unique NO, the condition (1) is sufficient by CMT 1.


Condition(2) tells us that M < 0 - This means N has to less than 0

=> For M = -1 and N = -1 => \(\frac{M}{N}\) = -1/-1 = 1 and \({MN}^4\) = (-1)^4 = 1 - \(\frac{M}{N} > {MN}^4\)- NO

=> For M = -4 and N = -2 => \(\frac{M}{N}\) = -4/-2 = 2 and \({MN}^4\) = 8^4 - \(\frac{M}{N} > {MN}^4\)- NO

Since the answer is a unique NO, the condition(2) alone is sufficient by CMT 1.

Each condition alone is sufficient.

So, D is the correct answer.

Answer: A

TIP: If con(1) = con(2), then 95% times D is the correct answer.

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.