What is the remainder when the positive integer n is divided by 12? GMAT Explanation

What is the remainder when the positive integer n is divided by 12?

(1) When n is divided by 6, the remainder is 1.
(2) When n is divided by 12, the remainder is greater than 5.

 

Define the question: What is the remainder when the positive integer n is divided by 12?

Not too much to note here other than we are looking only for the remainder here, not necessarily what n is. If every choice of n gives the same remainder, we still have sufficient information.

 

Organize the information: What do we know about n and the remainder?

Well, we know n is a positive integer, but not much else without the additional statements.

Let’s think about the remainder. Since we have a positive integer divided by 12, we know the remainder has to be an integer between 0 and 11.

Knowing this, let’s take a look at the first statement.

 

(1) When n is divided by 6, the remainder is 1.

This statement gives us some good information about n. Let’s break down what’s being said here. If n divided by 6 has a remainder of 1, this means n is 1 greater than a multiple of 6.

Let’s write down some possible choices of n to make this more clear:

1 = (6*0) + 1

7 = (6*1) + 1

13 = (6*2) + 1

19 = (6*3) + 1

25 = (6*4) + 1

etc.

 

Here, we have choices 1, 7, 13, 19, 25, etc. for n, and these all have a remainder of 1 when divided by 6. However, we need to find the remainder of n divided by 12. Let’s see what remainders these choices for n have when divided by 12:

1 = (0*12) + 1  (remainder = 1)

7 = (0*12) + 7  (remainder = 7)

13 = (1*12) + 1  (remainder = 1)

19 = (1*12) + 7  (remainder = 7)

25 = (2*12) + 1  (remainder = 1)

etc.

 

Looks like we have a pattern here. Based on possible values for n, our remainder when divided by 12 can either be 1 or 7. Since we can’t narrow this down further, statement 1 is insufficient.

Let’s look at statement 2.

 

(2) When n is divided by 12, the remainder is greater than 5.

This statement is referring to the remainder when n is divided by 12 – this is the remainder we are looking for!

Well, we already know this remainder has to be between 0 and 11. So this statement just further limits our options. If the remainder is greater than 5, that must mean it’s between 6 and 11.

However, we can’t narrow this down any more, so statement 2 is also insufficient.

 

Let’s try both statements together:

From statement 1, we know that our remainder is either 1 or 7. From statement 2, we know our remainder is between 6 and 11.

So, the only choice of remainder that satisfies both statements is 7, so that must be the answer. Both statements together were sufficient.

Correct Answer: C