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# If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is GMAT Explanation, Video Solution, and More Practice!

If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is

A. 6
B. 12
C. 24
D. 36
E. 48

We get this one a lot in GMAT tutoring sessions. Most people fail to understand what the questions is asking and then just start working without a plan. It ends up being that work itself isn't totally off but the student doesn't really know where to take it in order to narrow down the answer choices.

What also tends to confuse is that the question asks for the largest positive integer but we end up choosing the smallest possible value of n.

## Define the question

the largest positive integer that must divide n

There it is. But let's not leave it like that. Always try to do something with the questions. In this case because we're looking for something that divides evenly into n let's set up a fraction:

n/z = integer

And we want to maximize z because we're looking for the largest integer that must divide n.

The largest integer that divides any integer is itself. So really we're looking for n. Not that important to make that inference but just wanted to point it out.

## Setup

Now let's gather the information from the question and get things set up so we can make some inferences. We know that n is a positive integer and then n squared is divisible by 72. We can write out an equation with that second piece of information.

n^2/72 = integer

We're really looking to solve for n so let's go ahead and simplify this equation.

n^2 = integer*72

Take the square root of both sides.

n = √72√integer

Now let's pull out perfect squares from 72.

n = √9√4√2√integer

n = 6√2√integer

Now we can use the first piece of information that n is a positive integer. So 6√2√integer is a positive integer. Somehow the radicals have to disappear. So √2•√integer must be an integer. What's an easy way to do that? Make integer equal 2 so you have √2•√2 = 2.

n = 6*2 = 12

What's the largest positive integer that must divide n? 12

Now, you might be thinking: is 12 the only possibility for n? Or put another way, is 2 the only possible value for the integer? Good question. No. There are an infinite number of possible values for the integer and consequently for n. Any number that cancels out the radicals will work.

√2√8 = √2√2√4 = 4*6 = 24

√2√18 = √2√2√9 = 6*6 = 36

√2√32 = √2√2√16 = 8*6 = 48

Any number that has √2•perfect square will work.

So why is the integer 2 and the correct n 12? This is coming back to what tends to confuse GMAT tutoring students (looking for the largest integer that must divide but then the answer is actually the smallest possible value of n).

Because the question is a MUST. So you need the most basic building block of n. Look at all of the values we came up with for n: 12, 24, 36, 48. What's the biggest integer they have in common (least common multiple)? 12

Regardless of which multiple of n you come up with it will always be divisible by 12.

## Additional GMAT Divisibility Practice Question

Here's another divisibility question from GMAT Official Guide: If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following?

Here's a tricky exponents divisibility puzzle from GMAT question of the day

And another from the GMAT official guide that's not the same but has a similar puzzle/exponents/divisibility vibe with factorials in the mix: If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

# Last year the price per share of Stock X increased by k percent and the earnings per share of Stock X increased by m percent, where k is greater than m. By what percent did the ratio of price per share to earnings per share increase, in terms of k and m? GMAT Explanation, Video Solution, and More Practice!

Last year the price per share of Stock X increased by k percent and the earnings per share of Stock X increased by m percent, where k is greater than m. By what percent did the ratio of price per share to earnings per share increase, in terms of k and m?

A. k/m
B. (k–m)
C. 100(k–m)/(100+k)
D. 100(k–m)/(100+m)
E. 100(k–m)/(100+k+m)

This is a very challenging percent change word problem with a ton of variables from the GMAT Official Guide. A few things to remember when working on word problems:

2. Define the question before starting calculations (use the nouns to define the question you don't have to fill in the numbers)
3. Solve thoughtfully. Avoid simply burrowing in. Zoom in/Zoom out. Pause. Think.

Since there are variables in the answer choices this is a good candidate for: picking numbers.

Now, let's also remember the practical picking numbers test. If the question would be easier if you simply had the numbers then go ahead and pick them! This question would just be a percent change. So I'd say that is a clear signal to pick numbers.

Alright, we're getting a little ahead of ourselves. Let's go ahead and define the question: By what percent did the ratio of price per share to earnings per share increase, in terms of k and m?

That's it, no more no less. In tutoring sessions students often struggle getting this question defined. The question is always just that last line and doesn't need interpretation. If you're having trouble defining it just read the last line to yourself word by word verbatim. Often, if you're confused it's because you missed a word or added a word or two. Avoid paraphrasing here.

What gets missed on this one? That you're comparing RATIOS. It's the percent change of the RATIO of price per share to earnings per share. It's not Price compared to Earnings. It's P/E last year compared to P/E now.

So let's put that information into the percent change formula:

(((New Price Per Share/New Earnings Per Share)/(Old Price Per Share/Old Earnings Per Share)) - 1)100. The general formula is ((New/Old) - 1)100

We have a bunch of variables. The New/Old price and earnings and k and m. Are there any constraints? k > m. But that's it. So you can pretty much pick whatever numbers you'd like.

Any thoughts on where to start? I'd start by picking the old price and earnings then k and m which we'll use to derive the new price/earnings.

Because we're dealing with percents let's use 100 because that makes things easy.

Can you make earning and price the same? Yes! Why not?

Old

Price 100

Earnings 100

Now let's pick an easy k and m. I'd go for 10 and 5. or 20 and 10. Something simple. Let's do 10 and 5.

New

Price 110

Earnings 105

Now let's put those numbers into our percent change formula.

((110/105)/(100)(100) - 1)

Notice that by making old price and earnings the same the denominator cancels. So after cancelling the denominator and reducing 110/105 we're left with:

(22/21 - 1 )100 = (22/21 - 21/21)100 = 100/21

So that's our percent change, 100/21. Now we need to do what we always do picking numbers: plug the numbers into the answer choices to yield our answer, 100/21.

If you follow our blog and have read our other GMAT explanations you probably know that we have a shortcut for this. We're going to use divisibility to avoid some of the calculations.

Let's look at the denominator of our answer: 21. So we know that the correct answer must have a denominator that is a multiple of 21. I'd break down 21 to primes, 7 and 3. I'd focus on three because it has easy divisibility rules. So we also know that the denominator must be a multiple of 3. So let's start eliminating answer choices based on that.

A. k/m

10/5 = 2. Not Div by 3.

B. (k–m)

10-5 = 5. Not Div by 3.

C. 100(k–m)/(100+k)

110. Not Div by 3.

D. 100(k–m)/(100+m)

105 is Div by 3 so this is possible.

E. 100(k–m)/(100+k+m)

115 is not Div by 3.

## More Challenging GMAT Word Problem Practice Questions

Here's an almost identical question from the GMAT Official Guide: Last Sunday a certain store sold copies of Newspaper A

Here's a work and rate example that's a little different but you try picking numbers and using the divisibility trick for eliminating answers: During a trip, Francine traveled x percent of the total distance

# If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following? GMAT Explanation

If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following?

A) 3
B) 6
C) 7
D) 8
E) 10

## Define the Question

x must be a multiple of which of the following? What does that actually mean? We need to figure out the basic divisibility properties of x and then use that to eliminate answer choices. Do you we need to solve for x? If we could that would be great but we don't need to and probably won't be able to.

## Setup

There are two nice ways to approach this question. There's a more practical setup and a more number properties approach. Let's go practical first.

We know that Y is a multiple of 5. So why not pick some values for Y? Then we can plug those into the equation and get some possibilities for x and see what inferences we can make.

Let's start at the minimum y multiple of 5 value (considering that both x and y are positive integers), 5, and go from there, 10, 15, 20...

We probably don't more than that.

## Solve

Now let's plug those y values into the equation: 3x + 4y = 200

3x + 20 = 200

3x = 180

x = 60

3x + 40 = 200

3x = 160

x = 160/3

3x + 60 = 200

3x = 140

x = 140/3

3x + 80 = 200

3x = 120

x = 40

Out of our 4 values of y only 5 and 20 are valid. The others produce non-integer values for x. Let's look at the results for 5 and 20 and see what inferences we can about the divisibility properties of x.

We've 40 and 60. So what's common or the least common multiple? They both have a 5 and a 4. The 3 in 60 and the 8 in 40 are not common so are not a factor that x is must have. Let's look at the answer choices and see what we can eliminate.

A) 3 Not a must have because it's not a factor of 40
B) 6 Not a must have because it's not a factor of 40
C) 7 Not a must have because it's not a factor of 40 or 60
D) 8 Not a must have because it's not a factor of 60
E) 10 With this method we can't prove that this is a must have but we've eliminated everything else and it is common between 40 and 60.

## Setup 2

The other way to do this is to simplify the equation and then make some inferences.

3x + 4y = 200

3x = 200 - 4y

3x = 4(50 - y)

## Solve 2

Keep in mind that y is a multiple of 5. So again could be 5, 10, 15, 20... What happens when you put any of those numbers in there? 50 - 5 = 45. 50 - 10 = 40. 50 - 15 = 35. You always end up with a multiple of 5. That's because of this divisibility rule:

Div by a number + Div by a number = Remain Div by that number

Div by a number + Not Div by that number = Not Div by that number

Not Div by a number + Not Div by a number = Depends on the number

It's a good rule to know.

So 3x is equal to a multiple of 5 times 4. Meaning: 3x is a multiple of 5 times 4. What about x? Well, the 3 isn't a multiple of 5 or 4 so x must be a multiple of 5 and 4. E is the only option that matches. You might say: but E (10) only has a 2 not a 4! That's OK. We're not being asked: which of the following represents all of the factors of x. We're just asked which answer choice represents factors that x must have. And x must have a 4 and a 5. So it must have a 2 and a 5.

## Additional Number Property Divisibility GMAT Practice Questions

Here's another GMAT Number Properties Puzzle question that can be cracked with basic organization: The number 75 can be written as the sum of the squares of 3 different positive integers

And here's a number properties puzzle from GMAT Question of the Day that also benefits from just getting the info from the question in shape.

# If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p? GMAT Explanation

If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

A. 10

B. 12

C. 14

D. 16

E. 18

Here's a GMAT factorials puzzle from the GMAT Official Guide. I always tell GMAT tutoring students that this is a fantastic questions to get on a test, even though, at least initially, most people get it wrong. Why is it a good one? Once you know how to solve this puppy:

1. It's pretty easy to recognize it again
2. It's relatively quick to solve

Let's break this down.

p is the product of the integers from 1 to 30. This just means 30!. What's 30!? It's a the product of the integers from 1 to 30. Oh. Yeah. 30*29*28*27......*3*2*1

3^k is a factor of p. This means that 30! must be divisible by 3 raised to the k power. So 30!/3^k = integer.

We want to find the GREATEST integer k. Clearly you could make k = 1 and that would work as 30!/3 is an integer. But, again, we want the greatest k. So the question boils down to, how many 3's are in 30!?

There are two ways to do this. Let's start with the slow but practical way.

List out the components of 30! that are divisible by 3 and then count up all of your 3's. 30, 27, 24, 21, 18, 15, 12, 9, 6, 3. Keep in mind that 27 has 3x3's, 18 has 2x3's, and 9 has 2x3's. So you end up with 14 3's. Not bad.

Here's the faster/cleaner way to approach these factorial questions. Take the factorial and divide it by the number you're testing, in this case 3.

30/3 = 10. That means that there are 10 numbers from 1-30 that are divisible by 3. Done! Nope. Not yet because there are some numbers from 1-30 that have more than one 3 (see above list).

30/9 = 3. Next, up the power of 3 so that we're dividing by 9. This tells us that there are three numbers from 1-30 that have two 3's. Great - that was easy! Well, not quite there yet because there are numbers from 1-30 that have three 3's.

30/27 = 1. Up the power once more so that we're dividing by 27. That counts the number of 27's from 1-30 or the number of numbers with three 3's. There's just one, 27.

30/81 = doesn't fit. 81 doesn't fit into 30 so that tells us that there are no numbers from 1-30 that have four 3's. That's where you stop.

Then just add those, 10 + 3 + 1 = 14.

The explanation for this second method is much longer BUT the method itself tends to be much quicker than the manual calculation.

## Additional GMAT Factorial Puzzle Practice Questions

Not the same factorial setup but a solid GMAT Question of the Day puzzle example to get you thinking in a GMAT kind of way

Here's another GMAT Question of the Day to sharpen your GMAT puzzle skills.

# Club X has more than 10 but fewer than 40 members. Sometimes the members sit at tables with 3 members at one table and 4 members at each of the other tables, and sometimes they sit at tables with 3 members at one table and 5 members at each of the other tables. If they sit at tables with 6 members at each table except one and fewer than 6 members at that one table, how many members will be at the table that has fewer than 6 members? GMAT Explanation

Club X has more than 10 but fewer than 40 members. Sometimes the members sit at tables with 3 members at one table and 4 members at each of the other tables, and sometimes they sit at tables with 3 members at one table and 5 members at each of the other tables. If they sit at tables with 6 members at each table except one and fewer than 6 members at that one table, how many members will be at the table that has fewer than 6 members?

(A) 1

(B) 2

(C) 3

(D) 4

(E) 5

This is a tough question from the GMAT Official Guide that we almost always use as an example in our divisibility/remainder lesson. Without any setup most GMAT tutoring students get lost. They start out trying to do some sort of algebra and then fizzle out. Good news: there's a very easy way to tackle these (even though this style of question is considered difficult).

So let's start by reading carefully and defining the question: How many members will be at the table that has fewer than 6 members? Translate: what's the remainder when the total number of people is divided by 6?

So somehow we need to figure out the total number of members in Club X. What else do we have:

Sometimes the members sit at tables with 3 members at one table and 4 members at each of the other tables: When the total is divided by 4 the remainder is three.

Sometimes they sit at tables with 3 members at one table and 5 members at each of the other tables: When the total is divided by 5 the remainder is also three.

So now we have enough information to pinpoint the total. We know that it's a number between 10 and 40 which when divided by 4 or 5 has reminder 3. One practical way to approach this is to make a list of numbers that are remainder 3 when divided by 4 and another list that is remainder 3 when divided by 5. Then see at what number between 10 and 40 is in both lists.

Remainder 3/Divided by 4: 3, 7, 11, 15, 19, 23, 27, 31, 35, 39

Remainder 3/Divided by 5: 3, 8, 13, 18, 23, 28, 33, 38

The overlap is 23. So there must be a total of 23 members in Club X. 23 divided by 6 has a remainder 5.

How many members will be at the table that has fewer than 6 members? There will be 5 members at the table that has fewer than 6 members.

Making a list and cross-referencing is a great way to do these divisibility/remainder questions with members divided amongst tables, children assigned to classrooms, marching bands divided into rows... There is a much faster way though.

Take the least common multiple of the two numbers that you're dividing by, in this case 4 and 5, and then add the remainder, 3. So the LCM of 4 and 5 is 20. Add 3. 23. For numbers that don't share factors the LCM is just their product. Easy right?

These questions with people/things being divided into different equal numbered subgroups with some number left over often look challenging because there's usually a big block of text but, provided that you understand the above, are actually pretty simple and can be answered very quickly with very little calculation. These are great to get on an exam.

## Additional Divisibility/Remainder GMAT Practice Questions

You see these done both ways, with a story or more broken down. This example is spot-on content-wise but a more broken down example: GMAT Question of the Day Divisibility/LCM/Reminder Example

Here's another word problem divisibility question from Question of the Day for which you need to do some step by step practical work

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