Question: Which of the following is a factor of 18! + 1?

1. 15
2. 17
3. 19
4. 33
5. 39

Correct Answer: C
Solution and Explanation:
Approach Solution 1:

The problem statement asks to find the factor of 18! + 1.

We know that 18! and 18!+1 are consecutive integers.
We also know that two consecutive integers are always co-prime. This implies that they do not have any common factor other than 1.
For instance, 20 and 21 are consecutive integers, thus only common factor they share is 1.

Now, we can factor out each 15, 17, 33 = 3*11, and 39 = 3*13 out of 18!.
Therefore, 15, 17, 33 and 39 are regarded as factors of 18! and are NOT factors of 18!+1.
Hence, only 19 could be a factor of 18!+1.

Approach Solution 2:

The problem statement asks to find the factor of 18! + 1.
Let’s find the problem by analysing the options:

Option A: 15 - this can be written as 3*5. Therefore 15 is a factor of 18! since 18! = 1 * 2 * 3 * 4 * 5 * … * 17 * 18.
Since 18! + 1 and 18! are co-prime, 15 is not a factor of 18! + 1. Hence, option A gets eliminated.

Option B: 17 - Eliminated since 17 is a factor of 18! = 1 * 2 * 3 * 4 * 5 * … * 17 * 18.
Since 18! + 1 and 18! are co-prime, 17 is not a factor of 18! + 1.

Option C: 19 - Correct since 19 is a prime number. Therefore by applying Wilson's theorem, we get: 19 is a factor of (19-1)! + 1 = 18! +1.

Option D: 33 - this can be written as 3*11. Therefore 33 is a factor of 18! since 18! = 1 * 2 * 3 * 4 * 5 * … * 17 * 18.
Since 18! + 1 and 18! are co-prime, 33 is not a factor of 18! + 1. Hence, option D gets eliminated.

Option E: 39 - this can be written as 3*13. Therefore 39 is a factor of 18! since 18! = 1 * 2 * 3 * 4 * 5 * … * 17 * 18.
Since 18! + 1 and 18! are co-prime, 39 is not a factor of 18! + 1. Hence, option E gets eliminated.

Therefore, only (option C) 19 could be a factor of 18!+1.

Approach Solution 3:

The problem statement asks to find the factor of 18! + 1.

It is required to remember an important concept that cites:
If k>1 and k is a factor of N, then k is not the factor of N + 1.
For instance: since 7 is a factor of 350, 7 cannot be a factor of 350 + 1.

Another important concept that is required to remember is:
If N is divisible by k, then k is concealing within the prime factorisation of N.
For instance: 24 is divisive by 3 since 24 = (2)(2)(2)(3)

Now, let’s focus on the question:
The problem can be solved by using these two concepts. Let’s eliminate 4 of the 5 answer choices.

A: 15
Since 18! = (18)(17)(16)(15)....(3)(2)(1), we can see that 15 is a factor (divisor) of 18!
Therefore, as per the 1st Concept, 15 is NOT a factor of 18! + 1
ELIMINATE A

B: 17
Since 18! = (18)(17)(16)....(3)(2)(1), we can see that 17 is a factor (divisor) of 18!
Therefore, as per the 1st Concept, 17 is NOT a divisor of 18! + 1
ELIMINATE B

C: 19
Let’s leave for now.

D: 33
Since 18! = (18)(17)(16)..(11)..(3)(2)(1), we can see that 33 is a factor (divisor) of 18!
Therefore, as per the 1st Concept, 33 is NOT a divisor of 18! + 1
ELIMINATE D

E: 39
Since 18! = (18)(17)(16)..(13)..(3)(2)(1), we can see that 39 is a factor (divisor) of 18!
Therefore, as per the 1st Concept, 39 is NOT a divisor of 18! + 1
ELIMINATE E

Therefore, by the process of elimination, we can consider (option C) 19 as a factor of 18! + 1.

“Which of the following is a factor of 18! + 1”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide 2022”. To solve the GMAT Problem Solving questions, the candidates must have a basic knowledge of calculation and mathematics. The candidates can follow the GMAT Quant practice papers to go through varieties of questions that will enable them to improve their mathematical learning.

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