# If n is a positive integer and n^2 is divisible by 72

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? 