If n = 3^8 – 2^8, Which of the Following is Not a Factor of n? GMAT Problem Solving

Question: If n = \(3^8 – 2^8\), which of the following is not a factor of n?

1. 97
2. 65
3. 35
4. 13
5. 5

Correct Answer: C

Solution and Explanation:
Approach Solution 1:
The problem statement asks if n = \(3^8 – 2^8\), find which is not a factor of n among the options.

Let's solve this problem with an algebraic approach.

The first point we need to understand is that we're being assessed on the "difference of squares" algebraic factoring technique. Remember that the difference between squares has the following general form:

\(a^2– b^2 = (a + b ) (a - b)\)

Similarly, we can treat \( 3^8 – 2^8\) as a difference of squares, which can be expressed as:

\(n = (3^4 + 2^4)(3^4 – 2^4)\)

It can be further factored into \(3^4 – 2^4\) as an additional difference of squares, which can be written as:

\((3^2 + 2^2)(3^2 – 2^2)\)

This will give us:

\(n = 3^8 – 2^8 = (3^4 + 2^4)(3^2 + 2^2)(3^2 – 2^2)\)

Calculating it further below :

n = (81 + 16)(9 + 4)(9 – 4)

n = 97 * 13 * 5

It is asked if the following answers are NOT a factor of n, which we found to be equal to the product (97* 13* 5).

Now all we have to do is discover the answer option that does not divide evenly with 97* 13* 5
As visible, 97, 13 and 5 are all factors of 97* 13* 5.
We are left with 65 and 35.
On further notice, it is seen that 97* 13* 5 = 97* 65.
Thus, 65 also is a factor of n.
Only 35 is not.
Therefore the answer is 35.

Approach Solution 2:
The problem statement asks if n = \(3^8 – 2^8\), find which is not a factor of n among the options.

It is possible to solve the following equation and the arithmetic approach can also be useful in this case. This can be assessed in the following way:

\(n = 3^8 − 2^8\)

\(= (3^4 + 2^4) ∗ (3^4 − 2^4)\)

\(= (3^4 + 2^4) (3^2 + 2^2) (3^2 − 2^2)\)

\(= ( 3^4 + 2^4) (3^2 + 2^2) ( 3 + 2 ) ( 3 − 2 )\)

= 97 x 13 x 5 x 1

35 is not a factor of n since there is no 7 in n.
Therefore the answer is 35.

Approach Solution 3:
The problem statement asks if n = \(3^8 – 2^8\), find which is not a factor of n among the options.

n=3^8−2^8
=(3^4)^2−(2^4)^2
=81^2−16^2
=(81+16)(81−16)
=97×65

So it is clear that 35 is not the factor of 65×97

Therefore the answer is 35.

“If n = \(3^8 – 2^8\), which of the following is not a factor of n?”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide Quantitative Review 2022”. The candidate must possess concrete mathematical knowledge in order to solve GMAT Problem Solving questions. The candidates can go through GMAT Quant practice papers to analyse several types of questions to enhance their mathematical skills.

Suggested GMAT Problem Solving Questions

• If 2^x + 2^y = x^2 + y^2, Where x and y are Nonnegative GMAT Problem Solving
• If 3 < x < 100, for how many values of x is x/3 the square GMAT Problem Solving
• Hot Competition 3 Sep/8AM: In A Triangle ABC, Shown Above, Three Lines GMAT Problem Solving
• A Thief Steals a Car at 2.30 p.m. and Drives it at 60 km/h. The Theft GMAT Problem Solving
• If |x + 1 | > 2x - 1, Which Of The Following Represents The Correct Range GMAT Problem Solving
• Which Of The Following Is A True Statement? GMAT Problem Solving
• A machine puts c caps on bottles in m minutes. How many hours will it GMAT Problem Solving
• What is the least common multiple of 15, 18, and 24? GMAT Problem Solving
• The Price of Darjeeling Tea (in rupees per kilogram) is 100+0.10n GMAT Problem Solving
• A 5-meter-long wire is cut into two pieces. If the longer piece GMAT Problem Solving
• The average weight of 15 items is 8 kg. The least average GMAT Problem Solving
• In a class of 78 students 41 are taking French, 22 are taking German GMAT Problem Solving
• If |x| > 3, which if the following must be true? GMAT Problem Solving
• 100 students appeared for two examinations. 60 passed the first GMAT Problem Solving
• An integer between 1 and 300, inclusive, is chosen at random GMAT Problem Solving
• In How Many Ways Can 6 Chocolates Be Distributed Among 3 Children GMAT Problem Solving
• A Man Can Row 50 Km Upstream And 72 Km Downstream In 9 Hours GMAT Problem Solving
• A Sum Of Money Doubles Itself In 7 Years. In How Many Years It Becomes Four Fold GMAT Problem Solving
• On Selling A Pen At 5% Loss And A Book At 15% Gain, Karim Gains Rs. 7 GMAT Problem Solving
• If Two Typists Can Type Two Pages in Two Minutes, How Many GMAT Problem Solving
• An Empty Pool being Filled with Water at a Constant Rate takes 8 hours GMAT Problem Solving
• For Every Positive Even Integer n, the Function h(n) is Defined to be GMAT Problem Solving
• The formula F = 9/5 * C + 32 gives the relationship between the temperature GMAT Problem Solving
• A Takes 5 More Days Than B To Do A Certain Job And 9 Days More Than C GMAT Problem Solving
• A Watch Which Gains 5 Seconds In 3 Minutes Was Set Right At 7 A.M. GMAT Problem Solving
• Of The 800 Employees Of Company X, 70 Percent Have Been With The Company GMAT Problem Solving
• To 100 Litres Of Milk, 10 Litres Of Water Is Added And Then 20 Litres GMAT Problem Solving
• Walking at 3/4 of his normal speed, Mike is 16 minutes late GMAT Problem Solving
• There are 12 yes or no questions. How many ways can these be answered? GMAT Problem Solving
• There are 7 Red and 5 Blue Marbles in a Jar GMAT Problem Solving
• A Basket Contains 3 White and 5 Blue Balls GMAT Problem Solving
• Working together, John and Jack can type 20 pages in one hour GMAT Problem Solving
• Is x^2 *x^5*z>0? GMAT Problem Solving
• The average (arithmetic mean) of four distinct positive integers is 10 GMAT Problem Solving
• How many roots does the equation √x2+1+√x2+2=2 have? GMAT Problem Solving
• A Box Contains 10 Tablets of Medicine A and 15 Tablets of Medicine B GMAT Problem Solving
• If x=√10+3√9+4√8+5√7+6√6+7√5+8√4+9√3+10√2, then which of the following must be true? GMAT Problem Solving
• What is the last digit of (3)^(3)^3? GMAT Problem Solving
• It takes Jack 2 more hours than Tom to type 20 pages GMAT Problem Solving
• The angles in a triangle are x, 3x, 5x degrees GMAT Problem Solving
• If among 5 children, there are 2 siblings; in how many ways can the children be seated GMAT Problem Solving
• Among 200 People, 56% like Strawberry Jam, 44% like Apple Jam, and 40% like Raspberry Jam GMAT Problem Solving
• If 4 Women and 6 Men Work in the Accounting Department GMAT Problem Solving
• If 2^98=256L+N, where L and N are integers and 0≤N≤4, what is the value of N? GMAT Problem Solving
• A draining pipe can empty a pool in 4 hours GMAT Problem Solving
• If equation |x| + |y| = 5 encloses a certain region on the graph, what is the area of that region? GMAT Problem Solving
• Kate and David each have $10 GMAT Problem Solving
• How many five digit numbers can be formed using the digits 0, 1, 2, 3, 4, and 5 which are divisible by 3, without repeating the digits? GMAT Problem Solving
• Which of the following lines are parallel to line x = 4 – 2y? GMAT Problem Solving
• John has 12 clients and he wants to use color coding to identify each client GMAT Problem Solving