2^16 – 1 is Divisible by GMAT Problem Solving

Question: 2^16 – 1 is divisible by:

(A) 11
(B) 13
(C) 17
(D) 19
(E) 21

Correct Answer: C
Solution and Explanation:
Approach Solution 1:
The problem statement asks to find the number divisible by 2^16 – 1.
We can write it as 2 ^ 16 = ( 2 ^ 8 )^2 – 1
2 ^ 16 – 1 = ( 2 ^ 8 + 1 ) ( 2 ^ 8 – 1 )
We know that 2 ^ 8 is 256 so
= ( 256 + 1 ) ( 256 – 1 )
= 257 * 255
Which is 65535 that is divisible by 17 that is
= 65535 / 17 which is equal to
= 3855
So, 2 ^ 16 -1 is divisible by 17.

Approach Solution 2:
2 ^ 16 – 1 which we can simplify and write it as ( a ^ 8 – a ^ 8 ) ( a ^ 8 + a ^ 8 )
Where a is equal to two to the power eight and the value of b can be written as one two the power eight that is a = 2 ^ 8 and b = 1 ^ 8
= ( a ^ 8 –1 ) ( a ^ 8 + 1 )
= ( 2 ^ 8 - 1 ) (2 ^ 8 + 1 ) . . . . . . . . . . . . . . ( 1 )
Now we can substitute two to the power eight minus one as two to the power four minus one and two to the power four plus one which is
= ( a ^ 4 – 1) ( a ^ + 1 ) ( a ^ 8 + 1 )
= ( 2 ^ 4 – 1 ) ( 2 ^ 4 + 1 ) (2 ^ 8 + 1 ) . . . . . . . . . . . . . . . . ( 2 )
Two to the power 4 which is equal to 16.
Substitute it in equation ( 2 )
= ( 16 – 1 ) ( 16 + 1) ( 2 ^ 8 + 1 )
= 15 * 17 ( 2 ^ 8 + 1 )
Which is divisible by 17.
So, 2 ^ 16 -1 is divisible by 17.

Approach Solution 3:
The problem statement asks to find the number divisible by 2^16 – 1.
Two to the power 16 minus 1 is equal to 65,535 now will divide each option and check
A ) 65532 divided by 11 which is equal to 5,957.72 i.e 65532 / 11 = 5,957.72 which cannot be divided exactly
B ) 65532 divided by 13 which is equal to 5,041.15 i.e 65532 / 13 = 5,041.15 which cannot be divided exactly by 13
C ) now will see if 65532 is divisible by 17 i.e 65532 / 17 = 3855 which can be divisible by 17
D ) now will check if we can divide it by 19 i.e 65532 / 19 = 3,449.21 which is not exactly divisible by 19
E ) will check if we can divide 65532 by 21 i.e 65532 / 21 = 3,120.71. This is not exactly divisible by 21
So, it is clear 2 ^ 16 -1 is divisible by 17.

Approach Solution 4:
The problem statement asks to find the number divisible by 2^16 – 1.
We can write 2^16 – 1 as 16 ^ 4 – 1 ^ 4
We know that ( a ^ n – b ^ n ) is divisible by ( a – b ) ( a + b )
If n is even we can write it as 16 ^ 4 – 1 ^ 4 will always be divisible by ( 16 -1 ) and ( 16 + 1) and Hence the answer is 17
So, 2 ^ 16 -1 is divisible by 17.

“2^16 – 1 is divisible by”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide 2021”. To solve the GMAT Problem Solving questions, the candidates must have the basic concept of mathematics. The candidates can explore the GMAT Quant practice papers to practise varieties of questions that will enable them to improve their mathematical knowledge.

Suggested GMAT Problem Solving Questions

• 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
• Which of the following expressions has the greatest values? GMAT Problem Solving
• If @ x=x^2/2x^2-2 , What is the Units Digit of @ (@4)? GMAT Problem Solving
• What is the product of all possible solutions of the equation |x+2|- 5|x+2| = -6? GMAT Problem Solving
• If 69% of k is 23, approximately how much is 23% of k? GMAT Problem Solving
• Metropolis Corporation has 4 Shareholders GMAT Problem Solving
• If x and y are positive integers and (1/7)^x * (1/8)^12 = (1/8)^1/18y, then what is the value of y-x? GMAT Problem Solving
• At Springfield High, three-fourths of the male students and half of the female students speak a foreign language GMAT Problem Solving
• What is the number of integers from 1 to 1000, inclusive that are not divisible by 11 or by 35? GMAT Problem Solving
• If m is Three Times n, and if 2n + 3 is 20% of 25, What is the value of m? GMAT Problem Solving