Which of the Following is a Factor of 18! + 1? GMAT Problem Solving

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.

Suggested GMAT Problem Solving Samples

• The Length of the Diagonal of Square S as well as the Lengths of the Diagonals GMAT Problem Solving
• For a Certain Set of Numbers, If x is in the Set, Then Both –x^2 and −x^3 GMAT Problem Solving
• A Team Contributes Total of $399 From its Members GMAT Problem Solving
• Joe drives 120 miles at 60 miles per hour, and then he drives GMAT Problem Solving
• What is the probability of getting two cards that belong to the same GMAT Problem Solving
• From the Even Numbers Between 1 and 9, Two Different Even Numbers GMAT Problem Solving
• Three Workers, A, B, and C, can Complete a Certain Task GMAT Problem Solving
• A Certain Basket Contains 10 Apples, 7 of Which are Red and 3 are Green GMAT Problem Solving
• Which of the following equations has 1 + √2 as one of its roots? GMAT Problem Solving
• After distributing the sweets equally among 25 children, 8 sweets rem GMAT Problem Solving
• By walking at 4/5 th of his usual speed, Brent reaches his office 10 minute GMAT Problem Solving
• The Greatest Number of Four Digits Which is Divisible by 15, 25, 40 And 75 GMAT Problem Solving
• A Rack has 5 Different Pairs of Shoes. The Number of Ways in Which 4 GMAT Problem Solving
• What is the Number of Positive Divisors of the Form (4n + 2) of Integer GMAT Problem Solving
• In the Figure Above, the Area of Square Region PRTV is 81, and the GMAT Problem Solving
• Of the Following Expressions, Which One has the Smallest Value? GMAT Problem Solving
• A Dance Troupe has a Total of 50 Dancers Split into 2 Groups. GMAT Problem Solving
• At a Restaurant, Glasses are Stored in Two Different-Sized Boxes GMAT Problem Solving
• A Mixture of 70 Litres of Alcohol and Water Contains 10% of Water GMAT Problem Solving
• Let f be a function such that f(mn) = f(m)f(n) for every positive inte GMAT Problem Solving
• The metro service has a train going from A to B and from B to A every GMAT Problem Solving
• Let abcd be a general four-digit number and all the digits are non-zer GMAT Problem Solving
• Three-fourths of the area of a rectangular lawn 30 feet wide by 40 fee GMAT Problem Solving
• If x =(root 5- root 7)^2 , then the best approximation of x is: GMAT Problem Solving
• A briefcase lock is a combination of 4 digits where each digit varies GMAT Problem Solving
• A ladder of a fire truck is elevated to an angle of 60 degree GMAT Problem Solving
• A Television Manufacturer Produces 600 Units of a Certain Model Each GMAT Problem Solving
• A 5-Digit Code Consists of One Number Digit Chosen from 1, 2, 3 GMAT Problem Solving
• The Length of a Blackboard is 7 Cm More Than its Breadth GMAT Problem Solving
• The figure shows the shape of tunnel entrance. If the curved portion GMAT Problem Solving
• At a certain restaurant, a meal consists of one appetizer, one main co GMAT Problem Solving
• Last September the Sales total for a Certain Car Dealership was 400% GMAT Problem Solving
• When an Object is Dropped, the number of Feet N GMAT Problem Solving
• A Cube with an Edge Length of 6 Contains the Largest Possible Sphere GMAT Problem Solving
• Samantha's Investment Portfolio Consists of Three Stocks GMAT Problem Solving
• If 0 < x < 1, what is the median of the values x, x^-1, x^2 GMAT Problem Solving
• A Train Traveling at 72kmph Crosses a Platform in 30 seconds GMAT Problem Solving
• What Values of x have a Corresponding Value of y that Satisfies both xy > 0 GMAT Problem Solving
• A florist buys roses in bulk to be sold at a retail price of $30 per GMAT Problem Solving
• Find the Sum of the Sum of Even Divisors of 96 and the Sum of Odd Divisors GMAT Problem Solving
• A Bullet Train Leaves Kyoto for Tokyo Traveling 240 Miles Per Hour GMAT Problem Solving
• What is the Sum of 7 Terms of a 'geometric Progression' Whose First GMAT Problem Solving
• A Four-Character Password Consists of One Letter of the Alphabet GMAT Problem Solving
• A and B Started a Business With Initial Investments in the Ratio 5 : 9 GMAT Problem Solving
• A couple decides to have 4 children. If they succeed in having 4 child GMAT Problem Solving
• A Gambler began Playing Blackjack with $110 in Chips GMAT Problem Solving
• A Used Car Dealer Sold One Car at a Profit of 25 Percent GMAT Problem Solving
• If x – 1, x + 6, x + 7 are 3 side lengths of a triangle GMAT Problem Solving
• Find the next number of series: 3, 4, 8, 17, 33, 58. (a) 68 GMAT Problem Solving
• A pie is divided into 3 equal parts. The first part is divided equally GMAT Problem Solving