Real GMAT® Problems Ep. 10: Rates and Divisibility

Welcome to the GMAT Strategy Podcast. You're here because you believe there's a better way to study for the GMAT and so do we.

We created the GMAT Strategy to maximize your results and minimize your efforts so you can get to the fun parts about business school and life as quickly as possible.

My name is Isaac Puglia and I've been teaching GMAT classes and tutoring privately for the GMAT for almost a decade and I've achieved a 99th percentile score on the GMAT and helped thousands of students get into the business schools of their choice.

I'm excited to be a part of your MBA journey since we all at TGS think our world can benefit from the best possible business leaders that we can find.

If this show is bringing new value, please share it with your friends and family who are studying so that together we can make this process as easy and as painless as it can possibly be.

Let's go!

Welcome to Real GMAT Problems Episode 10.

Doing Real GMAT Problems is a great opportunity to get to know the exam better and also build your arsenal of strategies. So let's go through some more examples together and help you accelerate your progress toward your goals.

We're going to walk through three examples of actual past GMAT questions together. They're from the 11th edition of the official guide for GMAT review and you can web search them if you'd like to follow along visually but I know many of you will not be in that situation and that's completely fine.

I'll give you a chance to work through the solution on your own if you want to and then I'll discuss my thoughts on the problem and my recommendations for what to take away from each one.

What I recommend is pausing after I read through each problem and at the very least thinking about how you would solve the problem. You might not be able to execute all the steps in your head if you're on the go but see how far you can get with a mental solution. It's a great workout for your brain.

If you are on the go and have access to pen and paper like if you're on a plane or something like that, that's ideal. Try to make some progress on the problem before we review together and focus on good scratch work habits. You want what you write down to be clear enough that someone else can tell how you thought through the problem just by looking at what you write down.

If writing isn't practical right now because you're driving or running or at the gym, that's completely fine. Just do your best to visualize how you would write down your scratch work and try to mentally visualize it being that clear. What I was just talking about, use that as your metric for success. For most of us, this is going to be way more productive than trying to do all our computations mentally.

Here we go. This is problem number 31.

Problem 1: Juan Takes 11 Seconds to Run Y Yards

The problem says if Juan takes 11 seconds to run Y yards, like the variable Y yards, how many seconds will it take him to run X yards at the same rate?

Answer choice A is 11 times X divided by Y. B is 11 times Y divided by X. So A is 11X over Y, if you want to think about it that way, B is 11Y over X. C is X over 11Y. D is 11 over XY. And E is XY over 11. Take a moment and I'll jump in in a second.

So again, I'm not going to wait for any number of minutes here. Feel free to just pause the episode if you want to work. I'm just going to jump right into the solution so we don't wind up with a bunch of dead air.

So this is actually a great follow up to the rates question we left off within our most recent episode before this one, episode nine. And it's a great opportunity to use some more rows and columns to stay organized. If you want my take on how to stay really organized with word problems, then that problem is going to be at the beginning of the previous episode in the feed.

Like I mentioned last time, word problems often present more of an organizational challenge than a mathematical challenge. So it's a great idea to have some kind of system of visual organization. You don't need to use rows and columns specifically, but you want to have something that prevents you from missing questions you know how to do.

This question in particular is actually a great example of why that's really important.

What I like to do with rates questions specifically is set up the most common rates formula in my top row. I was talking about this last week. That common formula is rate times time equals distance. Then each item, rate, time and distance, becomes its own column header, and I can organize all the rates in the problem in column one, all the times in the problem in column two, and all the distances in the problem in column three.

This usually makes the problem easier to solve and it also makes it much less likely that I'm going to make a mistake and miss a question that I know how to do.

It's worth mentioning that there's no partial credit on the GMAT. You might already be aware of that but it's always good to be reminded. And that means things like your scratch work, like I talk about at the beginning of all these Real GMAT Problem episodes, it's so important. Because in a school test where there's partial credit, you might do the problem correctly all the way to the last step and then just mix up one number and you might get like a 9 out of 10 credit or an 8 out of 10 credit on that problem. But that's not going to happen on the GMAT.

If I make one tiny mistake and I get the wrong number at the end, it's a total zero. It's an all or nothing situation. And that's part of the reason you're going to hear me talk so much about these organizational approaches to problems, building good habits of visual organization, building good scratch work habits. Just because I've seen it be such a problem for so many super capable and super talented and super hard working people over the years that I just want to help you get ahead of the game on that. It's something I personally did not pay enough attention to at the beginning of my prep and it caused me a lot of pain on the back end. I was able to get things worked out and things have gone well since then, but I could have saved myself a lot of pain, a lot of time, a lot of energy, and I'd like to save that for you if at all possible.

Let's get back to the problem. The first piece of information I'm given is that Juan takes 11 seconds to run Y yards. I like to just fill that into the first row of the chart: under the T column I'm going to write 11 seconds and under the D column I'm going to write Y yards.

I'll also emphasize again how valuable writing the units of each quantity is on your paper when you're doing word problems. Highly recommend writing Y yards, not just Y, under the distance column. It's totally natural to feel resistance to that. A lot of people feel like I don't have time to write the units down or they're just too stressed and they have other excuses for not doing it.

So listen, at the end of the day I'm never going to argue with your results. If you're getting the results you want to see doing things the way you're doing them and it's different than what I'm recommending, great, don't change. But if you're not getting the results you want, then consider writing units on your word problems. It should help. Yes, it might feel slower, but I think you'll find that on average, at least, you will finish the problems faster even though you feel a little slower. That's what I've seen, at least.

What that shows me when I write that down is that I can solve for the rate, because I have two out of the three parts of the rate times time equals distance equation. Ask yourself, what would I need to multiply by 11 seconds to get Y yards? That's one way to do it. Or you can set up an equation like I was talking you through in the recent episode, with just R times 11 equals Y, and you can divide both sides by 11.

So that would get you Y over 11 as your rate: Y divided by 11, yards per second. I recommend writing that under the R in the rate times time equals distance formula. Here we're just filling out the chart with what's been given.

It's great that I just solved for the rate, because the question is asking how long it'll take Juan to run X yards at the same rate. Quote unquote, "at the same rate." That was a critical piece of the problem that I need to solve. Now I have that rate and I can get on with the solving of it.

If I'm asked to do another computation in a rates question, I like to just make another row in my chart and then fill in what I'm given as I go. In this case that would be X yards for the distance. They give me a different distance and ask me to do a new computation, so I'm just going to make a new row in the chart right underneath the row I just made, and I would write X yards for that second row.

That second row tells me he's running at the same rate, so I can just bring his rate down to that lower row that I just wrote down. You can see that that keeps me nice and visually organized. I can track where all the data points are going as I work through the problem. I'm much less likely to make a mistake, much more likely to see difficult and disparate connections, and I think you'll see that more and more as we get into harder and harder problems together or in your own prep. You may have seen them already.

So in that rate column I'm just going to write Y divided by 11, yards per second, again.

This is great news, because I want to know how long it's going to take him to run X yards at that same rate, so I'm in a perfect position to solve for that. I already have the rate and distance, so I'm just going to use the variable method. I'm going to put a T in to help me visualize the algebra. If you have a method that works better for you, that's more logical for you personally, then by all means feel free to use that. Again, I'm never going to argue with your results. This is just what I recommend and what I've seen work the most often for the widest variety of people, and we're talking many, many thousands of data points at this point.

What I have now is an equation that says Y over 11 yards per second, times T seconds, is equal to X yards. And that helps me see I just need to divide both sides of that equation by Y over 11 in order to get what I'm looking for.

Side note: if your equation skills are a little rusty, or your fraction division skills are a little rusty and you want a little brushing up there, we've got an episode on each of those topics, linked for you below.

So let's isolate T. Dividing by Y over 11 is the same as multiplying by 11 over Y, so let's do that to both sides of the equation. That gives us T on the left side of the equal sign and X times 11, all divided by Y, on the right side. So 11X over Y on the right side, which gets us to answer option A, which is the correct answer: 11X over Y.

So that's the algebraic solution, and it's very worth understanding. I think if you're not super comfortable with something like that, it's probably worth putting this problem aside and redoing it at least once a day for the next few days until you can get comfortable with that kind of setup and solution, because it really pays to be good at translating word problems into algebra and using algebraic techniques to solve questions.

It's not the only way to do this problem. I'm going to talk you through another way in a second, and you don't have to default to this way if it's not natural for you, but it's very worth rehearsing.

There's another solution, though, and some of you who tuned in last time might have been thinking about plugging in numbers, because you've got variables in the answer choices. Which is also a great move here. So let's talk through that.

I mentioned this in our most recent episode, but if you're not sure how to plug in numbers, well, we have an episode for you on how to do that, linked below. So feel free to check that out if you want a little brushing up on that as well.

But just quickly, to talk you through the basics of the steps I'm about to discuss: step one, you can pick any number that you want, as long as it fits the constraint of the problem, to replace the variables. And you can plug that number or those numbers into the problem and solve, and then you can plug those numbers into all five answer options and see if there's a match.

I'm not going to go deep on the technique here because there's a whole episode for you there. But let me demonstrate. Instead of Y yards, let's make Y equal to something that seems reasonably good in the problem, like a good number. I'm going to pick 22, because I can see there's like an 11 in there. I'm not really going to be able to pick an integer that divides 11 evenly, so at least I can pick an integer that 11 divides evenly, because chances are I'm going to be doing some multiplication and division in a rate times time equals distance problem. That's just how they typically go.

So let's make Y equal 22. I'm going to write that down, Y equals 22, and I'm going to put into my chart 22 yards. And we're going to say he ran 22 yards in 11 seconds.

For some of you, you're going to like this technique because it just makes it a lot more intuitive for you to solve for the rate at this point. But even if you're going to use this approach, I still recommend organizing it visually. It is just a great default behavior that I think you'll really find pays extremely good dividends over time, even if you don't need to do it 100% of the time. In fact, there are very few situations where this kind of visual organization is a bad idea, so I recommend using it 100% of the time. Very little downside, lot of upside.

If we map out our rows and columns with rate, time, distance across the top, we would have 22 yards under D on the right, 11 seconds under T on the left of that. Then we can write our yards per second to create an equation. All we have to do is divide 22 by 11, or divide both sides of the equation by 11, and we get 2 yards per second for our rate. You'll notice a lot of the same moves are going to be made here, we're just making them with integers instead of variables. And some people find these rates questions, and word problems in general, just much, much easier to think through that way. If that's not you, don't worry about the solution. If it is you, great.

Let's say we plug in 10 yards for X. We're going to need to plug in a number for X as well if we want to do all numbers. You can sometimes do a hybrid, where you plug in numbers for some and then do algebra for other variables. It's totally fine to do that if that's the kind of the flow that you're in. But in this case I'm just going to pick 10 yards for X. We can make a new row right below the first one, just like we did in the previous solution, put 10 yards in the distance column, and then we'll bring down the rate, 2 yards per second, in the rate column.

Then we can write T as a placeholder for time. Or, if the math is really straightforward for you, then you can just write the time in there, but I like to write the variable just to stay consistent. Consistent process yields consistent results. And I have struggled so much, so ridiculously much, with missing questions I know how to do, that I just refuse to let it happen anymore. And so I go to great lengths to prevent that.

We can then divide both sides of the equation by 2 to isolate T, and we would get 5 seconds as our answer.

Coming back to this plug in numbers approach: if you plug Y equals 22 and X equals 10 into all five answer options, then you would realize that only option A yields 5 seconds. We did 22 yards divided by 11 seconds to get 2 yards per second, then we plugged in 10 yards for X, and then we divided 10 yards by 2 yards per second to get 5 seconds. That's the answer to the question: it's going to take him 5 seconds to run 10 yards. Then we plug Y equals 22 and X equals 10 into all five answer options, and if you do the math there, feel free to pause and run the numbers, you'll realize that only option A yields 5 seconds.

Now you might notice that, again, we make a lot of the same moves when plugging in numbers. That's quite common. But again, just for certain types of people, it's much easier to think through the logic of the problem with actual numbers rather than variables. And if that's you, I recommend making the plug in number strategy your default strategy when you see questions with variables in the answers, and then just practice the algebra strategy as a backup. If you're the type of person who favors algebra, then by all means default to creating equations with variables. Just look at what gets you the best results personally.

What's interesting to note is that it's not so much about what approach you use, but how well you execute that approach that matters. I guess there's just an almost overwhelming number of parallels to that concept that we could make to other areas of life. So I guess it's worth noting. And my strongest recommendation here is to develop good habits of organizing your work.

No, I'm just like hammering this point, but it's because I care, so please forgive me if I'm just like the overprotective person here. But it's incredibly frustrating to invest so much time and energy into preparing for the exam and then miss a bunch of questions that you could have gotten right, like when it matters most, on test day. So my advice is just set yourself up with good habits from the beginning, and I believe you will greatly thank yourself for that later.

As always, if you have any questions about this problem or what I'm presenting here, feel free to DM us at The GMAT Strategy on current social channels, or you can email us at contact@thegmatstrategy.com, and we're more than happy to help.

Problem 2: The Lowest Positive Integer Divisible by 1 Through 7

Let's keep going, let's check out another problem. This one's quite a different genre. This is problem number 33.

The question reads: what is the lowest positive integer that is divisible by each of the integers 1 through 7, inclusive?

Option A is 420. Option B is 840. Option C is 1,260. Option D is 2,520. Option E is 5,040.

Take a moment and work this out if you have pen and paper. Otherwise, pause the episode and try to visualize where you would go in strategizing on a question like this.

On the surface it might just seem like the answer is one times two times three times four times five times six times seven, or what's properly called seven factorial. But that is actually not true. This question is not that simple.

Notice that this question is not asking for any positive integer that is divisible by each of 1, 2, 3, 4, 5, 6, 7. It's asking for the lowest. And that's a more difficult question to reason through, I think.

For most of us, answering a question like this correctly and consistently is going to require a bit of background with divisibility in general. So I'm going to talk through that. If you want some help understanding the basics of divisibility, then please check out our episode from our GMAT math basics series titled Divisibility, which is linked below in the description of this content.

I'm going to assume you have gone through that, though, and just jump into the way that I recommend approaching a question like this. And if you want to backtrack to that basics lesson, then please just pause the episode and do so at any point, and then pick back up here when you feel like you have that proper foundation.

So the key to most divisibility questions on the GMAT, in my opinion, is actually kind of simple, which is that you want to think about them as fractions.

Now that might just sound comically redundant or comically obvious, but hear me out on this. The vast majority of us have been dealing with fractions for a really long time, and so we're likely to have some very positive instincts that you can tap into in these questions. And funnily enough, I rarely see people visualize these divisibility questions as fractions, or they might set it up as a fraction but they do it in an unproductive way. So I'm going to try to help you avoid that.

As strange as this might sound, most of us do not usually think about divisibility in terms of fractions. So there may be some reconditioning that's required here, and that's totally normal. I went through it as well.

Let's start with what divisible means. It means that when x is divisible by y, and I divide x by y, I get an integer result. No fractions or decimals. This is what I go like deep and long on in that math basics series.

If you just think about that basic concept for a moment, you'll realize that that means everything in the denominator, if we set it up as a fraction, needs to cancel. Everything in the denominator needs to cancel with something in the numerator. Because if everything in the denominator doesn't cancel, we're not going to get an integer result out of that fraction. It'll just stay a fraction, because there's going to be stuff in the denominator that doesn't cancel out.

So if we want x to be divisible by y, we need to make sure that y cancels out when we write x over y as a fraction.

Now with smaller numbers this usually isn't a very big deal to figure out. For example, let's say I asked you for the biggest odd number that 6 is divisible by. You can probably reason that it's going to be 3. If we put 6 on top of a fraction and we need an odd number in the denominator that's going to cancel, it's probably not hard to see that 3 will cancel with 6 and you'll get 2. You probably could walk yourself through that logic in a fairly straightforward way.

But what if the numbers are bigger? That's where prime factoring can really, really help out.

We discussed prime factoring a bit last time, and if you want help understanding the basics of prime factoring, we've got a math basics lesson on that linked below as well. But essentially what you're doing when you're prime factoring is you're dividing by prime numbers repeatedly, and you can do this with any integer. I'm just going to assume a basic knowledge of prime factoring here. If you want help with that, then just pause the episode, go back to last week, go back to that prime factoring basics lesson, and then come back to this one, and I think you'll find this next explanation quite productive.

So let's return to our simpler example for a moment of 6 over 3. If you prime factor 6, you'll get 2 times 3. And if you write that as a fraction, 2 times 3 on top and 3 on the bottom, it should be straightforward to see that the 3s will cancel with each other.

One final aside: if you're not sure how that works, we've got a math basics episode on fractions and how to cancel common factors in fractions, linked below as well.

So the whole prime factoring thing is almost certainly overkill with 6 and 3. But let's try a more complex example. How about 1,500 divided by 20? That's definitely not the hardest example, but we'll just use it as a case study for how this technique works.

If you prime factor 1,500, you'll end up with 2 squared, times 3, times 5 cubed. If you want help with basic exponents, we've got that in the math basics series as well. If you prime factor 20, you'll end up with 2 squared times 5. So imagine those two numbers in a fraction, with 2 squared times 3 times 5 cubed on top, and 2 squared times 5 on the bottom. Will everything in the denominator cancel? Sure will. The two 2s are going to cancel with the two 2s in the numerator. The 5 is going to cancel with one of the 5s from 5 cubed. And we're going to wind up with an integer.

And in a lot of these GMAT divisibility questions, you don't even care what that integer is. That's particularly true on data sufficiency divisibility questions. But this is a question where we want some actual numbers, so it pays to have the prime factoring thing well rehearsed by test day.

So that's one way that you can tell that 1,500 is divisible by 20: when I divide 1,500 by 20, everything in the denominator cancels and I get an integer result.

Now most of the time, divisibility problems on the exam are going to be like the example I just walked you through, where they give you the numerator and they ask you to figure something out about the denominator, generally based on what prime factors are in the numerator. This is why I recommend visualizing divisibility questions as fractions and then using prime factors if the numbers are really big.

What's really interesting about the problem that we're looking at from the 11th edition, the one I read to you before I started going through this divisibility theory and process lesson, is that we are actually given the opposite situation. They tell us something about the denominator, it's got to be able to accommodate 1, 2, 3, 4, 5, 6, 7. And then they ask us to use that to figure out what the numerator should be.

Now the good news is we can use the exact same concept. We just need all the prime factors in the denominator to exist in the numerator. If that happens, if all the prime factors of the denominator exist in the numerator, then everything is going to cancel when we go to divide. So even though this problem is inverted, and typically you'd see it in the opposite order, it's not going to change much of the technique that we're going to use.

So I already mentioned that it would be a lot easier to just do 1 times 2 times 3 times 4 times 5 times 6 times 7. It'd be a lot simpler. And if you think about the prime factors involved in multiplying all those numbers, then the answer in the numerator would be quite easy. It's just 1, 2, 3, 4, 5, 6, 7 multiplied together. But again, the question asks us for the lowest, the lowest positive integer that's divisible by each of 1, 2, 3, 4, 5, 6, 7. So we're being asked for a number that would be divisible by each of those individually, not necessarily the product of all of them.

So we don't want every prime factor of 1 times 2 times 3 times 4 times 5 times 6 times 7 in the numerator. That's going to give us a number that's too big. We just want the minimum number of prime factors that satisfies the constraints of the problem.

So let's go one number at a time here. I'll walk you through how I would be thinking about this.

Let's start with 1. Every integer is divisible by 1, so we can kind of just ignore that. But for teaching purposes, and to make the rest of the concepts that I want you to take away from this problem clear, let's just put a 1 in the numerator. It's not crucial to solve the problem to actually do that, it's just there for continuity and clarity of the lesson.

What about making sure the number's divisible by 2, though? Well, if I just have a 1 in the numerator, that 2's not going to cancel with that. So we'll definitely need to put at least one 2 in the numerator so that when we divide by 2 in the denominator, that cancels out. So in the numerator right now I would have written 1 times 2.

But if we just have 1 times 2 in the numerator, we won't be able to get 3 to cancel out when we try to divide by 3. So let's write times 3 in the numerator as well. And you can kind of see where I'm going with this.

So in the numerator right now we've got 1 times 2 times 3.

Quick aside: the reason we want to multiply all the numbers we're putting into the numerator right now is so that the denominator can cancel out. If we start adding, 1 plus 2 plus 3 in the numerator instead of multiplying, we're going to have some terrible difficulty getting things to cancel. So with divisibility in general, you're going to want to do multiplication with prime factors, and prime factoring in general is very multiplication based. Again, go back to last week's lesson and the math basics lesson on canceling fractions and primes. That should help if you feel like you don't understand what I'm talking about right now.

So we've got 1 times 2 times 3 in the numerator, and that gives us a number that's divisible by 1, it's divisible by 2, and it's divisible by 3. So we're making good progress on finding a number that's divisible by 1, 2, 3, 4, 5, 6, 7.

Now we go to division by 4. Think about this for a second. What kind of prime factors do I need in the numerator to get 4 to cancel out in the denominator? Think about it like you're writing a question mark in the numerator of a fraction and then 4 in the denominator. If you think about prime factors here, hopefully you're going to be thinking that you need 2 times 2 in the numerator to get the 4 to cancel out in the denominator.

But note that we already have a 2 in the numerator from when we wanted to be able to cancel out 2. So we don't actually have to put 2 more 2s in the numerator. We don't have to multiply by 4. We can just multiply by 2 one more time, and then make the numerator 1 times 2 times 3 times 2. And then we'll have a number that is divisible by 1, 2, 3, and 4, because all we need when we divide by 4 is the 2 times 2 to cancel out.

Now I just want to reiterate that the question is not asking you for a number that's divisible by the product of 1 times 2 times 3 times 4. If that were the case, we would need to multiply by 4 in the numerator, or 2 squared. We wouldn't just be able to multiply by 2 one more time. But the question is not asking us that. It's asking us for a number that's divisible by each of the numbers 1, 2, 3, 4, 5, 6, 7. So as long as each one divides, one at a time, we're good. So that's why we only have two 2s in the numerator so far.

So let's move on to 5. 5 is prime, so this one is simple. We've got to have a 5 in the numerator. 5 won't be able to cancel with 1, 2, 2, 3, or anything like that. So let's multiply by 5 in the numerator, and we would have 1 times 2 times 2 times 3 times 5.

That brings us to 6. What would you need in the numerator in order for 6 to cancel out if you divide? Again, if this is a challenge to think through, just visualize it as a fraction: question mark on top of the fraction, 6 on the bottom of the fraction. What would you need in the numerator for everything in the denominator to cancel out? Hopefully you're thinking you need 2 times 3 in the numerator. Again, if you're not sure how this works or you want a little more solidity, we've got those episodes linked below.

Note that in this case we've already got 2 times 3 in the numerator, because we started with wanting to make the numerator divisible by 2 and also by 3. And because we are not trying to divide by 2, 3, and 6 all at the same time, we just need to ensure that the numerator is divisible by 2, 3, and 6 individually. So the 2 times 3 in the numerator that we already have already guarantees it's going to be divisible by 6. So there's no need to add anything additional for division by 6 to work.

And that brings us to the last leg here, which is 7. So like 2, 3, and 5, 7 is prime, so we simply need to multiply the existing numerator by 7 and we're going to be good to go with division by 7.

So what we've got in the numerator is 1 times 2 times 3 times 2 times 5 times 7. And that is the bare minimum that will yield an integer that, when we divide by each integer, 1 through 7, we're going to get each one of those numbers to cancel out.

So we're just going to do a little multiplication to find our final answer. 2 times 3 is 6. 6 times 2 is 12. 12 times 5 is 60. And 60 times 7 is 420. And that's option A.

So on the surface this looks rather simple to most folks, and it is rated as an easy difficulty level question. But I think to truly understand the optimal approach requires some fairly deep thought about how numbers are constructed with prime factors and what the nature of divisibility is. At least it required that for me. So hopefully I've helped you understand both of those concepts a little bit better.

Now, quick note about difficulty ratings. These are generated by exam data collection. When a question writer creates a question, that person doesn't necessarily know whether the question is easy, medium, or hard. Instead, they serve the question with a bunch of different answer choices to a large number of people, and they simply collect data on how many people at each scoring level get the question right versus wrong. They look at your final score, and they look at whether you got that question right or wrong, and they use that to calibrate the difficulty rating of the question.

So this should make a lot of sense why it's done that way. But if you're not sure why that's a reasonable way to construct the exam, reach out any time and we'll do our best to explain. Again, just at The GMAT Strategy, if you want to DM, and contact@thegmatstrategy.com if you want to email.

Now, just kind of running with this, though, I think one of the reasons this question has a low difficulty rating is that if I just test option A right away, like I kind of try to brute force it by just dividing option A by 1, 2, 3, 4, 5, 6, 7, I immediately get an answer that works, and it is the lowest of all of the options. And so I'm done. And that means I can kind of hack the problem without truly understanding all the theory I just walked you through, and I can still get it right.

Having said that, though, imagine dividing 2,520 by 2, 3, 4, 5, 6, 7, manually. It's probably going to take longer than the analysis we just went through above, once you are comfortable with the concepts I was just discussing. So it can take time to get comfortable with those concepts, but once you are comfortable with them, I think it'll create a lot of speed. And that concept of thinking about divisibility as a fraction with prime factors will scale really well to almost every divisibility question you'll see.

Now, there's a good point to make there. There is a little bit of an investment required up front to learn the system that I just talked you through. And that's going to be a much larger investment for most people than learning how to divide 420 by seven different numbers. So that is a little bit of a trade off that you're going to face with some of these questions.

And that kind of comes in a question of like, should I put in the time to learn the fast, scalable solution, which takes more time in the short run, or should I try to hack as many questions as possible?

And it might seem like the answer is obvious on the surface, but it's going to be really tough for me to give you advice about your specific situation in this broader format. And so there's actually no right or wrong answer to this. Some people should actually hack every problem they can and just divide 420 by a bunch of numbers and completely avoid theoretical approaches like the one I just talked through. And then some others of you should do the theoretical approach, sometimes hack, at other times. Others of you should always default to the theoretical approach.

And this is why I think a lot of students can get frustrated in the process, because they can't just say, tell me the best way to do it.

So let me try to be as helpful as possible here. Test each solution out on your own. If you really want to be great, just put in the time to try several solutions, or whatever available solutions there are, to a problem, and figure out how to solve problems in different ways, and develop a thesis about when exactly should you personally brute force a question and when should you personally take the time to learn a more elegant solution.

It's my personal belief that nobody will be able to answer that but you. So my advice is just do simple data collection on the problems you complete and make notes to yourself about what you think is going to be best for you in terms of solution next time you see a question like that, and make some notes about how you're going to recognize that type of question. What are the features about the question that make it a good fast solution one, or a good hack solution one?

If you start this now and you don't stop, you'll have built your own personal success system, quote unquote, by test day.

If you want to find multiple approaches to problems, try web searching them. You can also ask a generative AI if there are alternate solutions to the first one it presents to you. And of course, you can keep listening here, since clearly I'm a fan of presenting multiple ways of solving problems to you.

And those are all very solid options, and I recommend testing each one of those out and figuring out what works best for you personally. Some people are like big, big Gen AI nerds, and people are like, I hate Gen AI, it's like so buggy or whatever. And there's no judgment here, like just do what works for you.

As you've hopefully experienced now, I try to recommend certain solutions for certain types of people. But in this generalized format, it's just going to be tough for me to give every single one of you the specific right advice for you personally. There's a lot of variance there, in my experience.

So let me just give a little bit of quick advice here that will help everybody. There aren't always multiple solutions to questions, and that's an important point. So don't stress if you're not finding any, or nothing's compelling with alternate solutions. You might find a couple solutions that are kind of all the same, like different flavors of the same ultimate approach. So I wouldn't worry about that. You're just looking for really unique solution paths, like some of the ones I presented to you here, where it truly is a totally different way of solving the problem. And if you can find those, then figure out which one is going to work best for you. And note that it's a very simple process. I think you'll find that super beneficial if you engage with that over time, and you can do that with any program, any provider, free program, paid program, doesn't matter. You can always do that.

Now, real quick, a couple points on that. There are a lot of really opinionated people in the GMAT industry who will be happy to preach to you about why their solution is best and why every other solution is bad. I'm not here to judge that. If that's what you want, that's totally cool. Again, I'm never going to argue with your results. I'm just telling you my experience is that when people discover and play to their strengths, they tend to get faster results on the exam. This is just what I've seen over tens of thousands of data points at this point, if you can believe that.

Since my goal for you is for you to be done with this process with your goal score as fast as possible, that's what I recommend doing, and that's why I just made those recommendations. But as always, it is completely up to you how you proceed. And if you're that type of person who's like, just tell me the quote unquote best way, then you can just always default to the first way I solve the problem, and then you don't have to think about or consider multiple solutions. I've structured all these lessons with you specifically in mind if you are that type of person.

So I realized I said we were going to go through three problems at the top of this lesson, but I also just realized I am extremely pressed for time this week. So I'm actually going to leave the next problem for the next lesson. Time allowing, let's plan for four problems next episode.

But wow, time flies when you're having fun there, and apparently I've talked through like quite a bit of deep theory there. So let's just pause with two for this week's episode. And definitely give me feedback if you're like, hey, that sucks, you know, I was really hoping for three. I will do my best to manage my schedule better in the future. It's just one of those weeks over here.

So that's where I'm going to wrap for today's lesson. I hope you took at least one valuable learning away from listening. Ideally more. As always, I'm just going to reiterate, if you have questions or you're not sure how this applies or what to do in your personal situation, reach out to us anytime and we'll do our best to help you out.

We are at The GMAT Strategy on current social channels if you want to DM, and contact@thegmatstrategy.com if you prefer email. Also, as always, my greatest hope is that this material will make your studies as easy and as painless as they can possibly be. If you want more tips and strategies for optimizing your performance on the GMAT, please head to our website, thegmatstrategy.com, which is linked in the description of this content, and check out our free video on how you can reach your dream GMAT score in half the normal time.

In the meantime, this is a regular show, so please subscribe. And please stay positive and stay consistent with your studies, everybody. I'll talk to you all soon.