Which of the Following is an Integer? GMAT Problem Solving

Question: Which of the following is an integer?

I. 12! / 6!
II. 12! / 8!
III. 12! / 7!5!

I only
II only
III only
I and II only
I, II, and III

Correct Answer: E

Solution and Explanation:
Approach Solution 1:
The problem statement asks to find out the integer from the following statements.
To find if the following statements are integers, let's look at how factorials can be expanded and expressed.

We can use 6! as an example.
6! could be expressed as:
6!
6! x 5 x 4!
6! x 5 x 4 x 3!
6! x 5 x 4 x 3 x 2!
6! x 5 x 4 x 3 x 2 x 1!
Acknowledging how this factorial expansion works will assist us in working through each answer choice, particularly options 1 and 2.

Statement l is 12!/6!
Since we already know how factorials can be expanded, let us solve this further
12! = 12 x 11 x 10 x 9 x 8 x 7 x 6!
When we insert this into answer statement l, we get

\(\frac{(12 *11 * 10 * 9 * 8 * 7 * 6!)}{6!}\)
= 12 x 11 x 10 x 9 x 8 x 7, this is an integer.

Let's analyse statement ll which is 12!/8!
Once again, Since we already know how factorials can be expanded, let us solve this further
12! = 12 x 11 x 10 x 9 x 8!
When we insert this into answer statement ll, we get

\(\frac{(12 *11 * 10 * 9 * 8!)}{8!}\)
= 12 x 11 x 10 x 9, this is an integer.

Let's analyse statement lll which is 12!/(7!5!)
Once again, Since we already know how factorials can be expanded, let us solve this further

\(\frac{(12 *11 * 10 * 9 * 8 * 7!)}{7!5!}\)

\(\frac{(12 *11 * 10 * 9 * 8)}{5*4*3*2*1}\)
\(\frac{(12 *11 * 10 * 9 * 8)}{12*10*1}\)

11 x 9 x 8, this is an integer.

Therefore, it has been analysed that statements l, ll, and lll are all integers.

Approach Solution 2:
The problem statement asks to find out the integer from the following statements.

It is already known that 12! is a multiple of all numbers from 1 to 12, so dividing it by 6! or 8! almost certainly yields an integer.

According to the above statement, as you can see,
Statement l and statement ll are integers

Now let’s consider statement lll
This is a question for the Combination Formula.

If we rephrase the equation to the question "How many different combinations of 7 people can you form from a group of 12 people?". The number of groups in these types of "Combination" calculations is always an integer.

Therefore 12!/7!5! will simplify to an integer.

Hence, it has been analysed that statements l, ll, and lll are all integers.

Approach Solution 3:
The problem statement asks to find out the integer from the following statements.

For any positive integers m, n and p, we will get two cases:
1) If m > n, then m!/n! is always an integer.
2) If m = n + p, then m!/(n!p!) is always an integer (which can be derived as mCp).

From the above two points, we notice that all three statements in the Roman numerals must be integers.

Therefore, the following statements l, ll, lll are all integers.

“Which of the following is an integer” – is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Official Guide Quantitative Review”. The candidate can practice these GMAT Problem Solving questions in order to crack the GMAT exam. The candidates can follow the GMAT Quant practice papers to polish up their mathematical skills so that they could solve quantitative problems easily.

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Which of the Following is an Integer? GMAT Problem Solving

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