If n is an integer, what is the greatest common divisor of 12 and n ?

Don't fall for the trap of this question. It isn't about the math. Many explanations of Quantitative questions focus blindly on the math, but remember: the GMAT is a critical-thinking test. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time (giving you more time for harder questions.) The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. Ready? Let's talk strategy here. Here is the full "GMAT Jujitsu" for this question:

The first thing we need to understand is the concept of a greatest common divisor (often called the greatest common factor or \(GCF\).) The GCF is the largest factor that is shared between two integers. And yet, once we know this, the fundamental trap of this problem is getting you to think that you actually need to solve for the GCF, instead of stopping as soon as you know you CAN solve. Many people spend too much time on Data Sufficiency questions because they think they need to get to the bitter end. We don’t. As soon as we have enough information to conclude that a statement is either sufficient or insufficient, we can move on.

For example, let’s evaluate statement #1. It states that the “product of \(12\) and \(n\) is \(432\).” Immediately, you should recognize that you can solve for \(n\), since \(12n = 432\). One equation. One variable. No weirdness. No possibility of multiple values. You don’t have to do the long division here -- actually solving for n -- to determine that statement #1 is sufficient. After all, if you can solve for \(n\), you can obviously determine what the factors of \(n\) are, and thereby determine what the GCF of \(12\) and \(n\) are. Case closed. Statement #1 is sufficient. Notice that you don't even need to know the answer to the question "what is the greatest common divisor?"

Statement #2 is similarly deceptive, but in a different way. Here, you can’t solve for a specific value of \(n\). However, if the “greatest common factor of \(24\) and \(n\) is \(12\)”, then \(12\) is clearly a factor of \(n\). And we also know that \(12\) is the greatest factor of \(12\) (after all, you can’t have an integer factor of a number greater than the number itself.) So, no matter how you look at it, we know that the greatest common factor of \(n\) and \(12\) must be \(12\). Statement #2 is also sufficient.

With both the statements individually sufficient, the answer is D.

Now, let’s look back at this problem from the perspective of strategy. This problem can teach us several patterns seen throughout the GMAT. First, minimize your math. I have met many engineers who think the Quantitative portion of the test is all about the math, and they end up doing pretty poorly on the GMAT because they fail to realize the test is actually measuring something besides their math skills: critical-thinking. This is especially true with Data Sufficiency. You only need to do "sufficient" math to prove that there must be only one answer to the question. (And this problem requires almost zero math if you think about it.) Second, statements in Data Sufficiency questions often bait you into thinking that you solve each statement in the same way. But the GMAT rewards flexible-thinking, not linear-thinking. With Statement #1, we can solve for \(n\). With Statement #2, it is impossible to know what \(n\) is. But the question isn't asking what \(n\) is. It is basically asking, "is there enough information here to calculate the GCF of two numbers?" On GMAT questions, you must always focus on exactly what the actual question is asking. For this question, we can arrive at an answer by leveraging the information in the problem. And that is how you think like the GMAT.

Statement 1:
12n = 432.
Since we can solve for n, the GCF of 12 and n can be determined.
SUFFICIENT.

Statement 2:
24 = 2*2*2*3.
Since the GCF of n and 24 is the blue product (12) without the red factor (2), n must be a multiple of 12 that does NOT include an additional factor of 2.
In other words:
n = 12a, where a is an ODD positive integer.
Options for n:
12*1, 12*3, 12*5, 12*7, 12*9...
In every case, the GCF of n and 12 is the value in green (12).
SUFFICIENT.


_________________

GCD means the largest number which has both 12 and n as its multiple..
So you require to know the value of n or some relationship between 12 and n or some property of n..

1) 12 * n =432...
n can be found and GCD of 12 and n can also be found..
Sufficient

2) GCD of 24 and n is 12.
This tells us that n is multiple of 12...
Let n =12x, where X is an integer
So GCD of 12 and 12x has to be 12 itself
Sufficient

D
_________________

(1) n=36
gcd=12
(2) 12 divides n
gcd=12

It must be D. Each statement alone is sufficient

From Statement 1: "The product of 12 and n is 432" i.e. n=432/12= 36.
So Greatest common divisor of 12, 36 can be derived. - Sufficient

From Statement 2: "The greatest common divisor of 24 and n is 12.".
that is n could be, 12, 36, 60, 84.... and so on. And, in each case, the Greatest common divisor for 12 & n will be 12. - Sufficient.
D is the answer.

Solution

Given:

• n is an integer

To find:

• The greatest common divisor of 12 and n

Analysing Statement 1

• As per the information given in statement 1, the product of 12 and n is 432

o Therefore, we can find the value of n = \(\frac{432}{12}\)
• As we can find n, we can also find the greatest common divisor of 12 and n

Hence, statement 1 is sufficient to answer the question

Analysing Statement 2

• As per the information given in statement 2, the greatest common divisor of 24 and n is 12

o So, the number n must be a multiple of 12
o Therefore, the greatest common divisor of 12 and n will be 12

Hence, statement 2 is sufficient to answer the question

Hence, the correct answer is option D.

Answer: D
_________________

hey pushpitkc,

i dont get if the product of 12 and n is 432" it means n = 36

and the question asks: If n is an integer, what is the greatest common divisor of 12 and n ?

how can 36 be divisor of 12

Also From Statement 2: "The greatest common divisor of 24 and n is 12.".

i dont get how greatest common divisor of 24 can be 12, 36, 60, 84 ?

any ideas ?

thanks and have a great day

Hey dave13

For the first statement, it says which is the greatest divisor the two integers, n and 12.
The greatest common divisor (GCD) of a set of integers is the largest integer that divides
each integer in the set. So, if n=36, the greatest common divisor is 12.

As for the second statement, the number n and 24 have a greatest common divisor of 12.
So, n can be any multiple of 12 - 12,12*2 = 24,12*3 = 36, and so on. This link should
help you understand what the Greatest Common divisor is.

Hope this helps you!

We need to determine the greatest common divisor or the greatest common factor (GCF) of 12 and n. If we can determine the value of n, then we can determine the GCF of 12 and n.

Statement One Alone:

The product of 12 and n is 432

Since 12n = 432, n = 432/12 = 36. So the GCF of 12 and 36 is 12.

Statement one alone is sufficient.

Statement Two Alone:

The greatest common divisor of 24 and n is 12.

Since the GCF of 24 and n is 12, that means n itself is a multiple of 12. Therefore, the GCF of 12 and n must be also 12.

Statement two is sufficient.

Answer: D
_________________

given n is an integer

we need to find the HCF of 12 and n.
remember that we need to find the HCF, not the "n".

Statement 1 gives straight answer
1) 12 * n =432
n can be found and HCF of 12 and n can also be found.
there ends the matter
Sufficient

Statement 2 says
the HCF of 24 and n is 12.
means that n is a multiple of 12

Let n =12p, where p is an integer
So HCF of 12 and 12p has to be 12 itself
Sufficient

1) 12n = 432
n = 36
two numbers are 12, 36
So the greatest common divisor is 12
Statement 1 is Sufficient.

2) 12 is divisor of n and 24, then surely it will be a divisor of 12.
Statement 2 is sufficient.

Ans. D
_________________

Answer: Option D

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