GMATPREP ? -Subtracting Exponents and Prime Factors - Does this shortcut reall work?

[adiknish]

By the way, a very similar problem was on my REAL GMAT exam last month.

 

The problem is as follows.

What is the greatest prime factor of (4^17) - (2^28)?

 

A) 2

B) 3

C) 5

D) 7

E) 11

 

 

D

 

 

My short cut is below in the spoiler. Do you think it works for all cases?

 

 

Step 1:Create equal -> 4^17 - 2^28 is equivalent to 4^17 - 4^14

Step 2: Subtract the exponents; 17-14 =3

Step 3: Next, take an easier example to get to a 3 exponent difference still using the number 4 as your base.

Example: 4^4 - 4^1 = 256 - 4 = 252

Step 4: Now, working backwards from the answer choices, starting with (E), divide 252 by all answer choices and see what is the highest prime factor to divide into 252.

(E) 252/11 = fraction -> Not a prime factor

(D) 252/7 = 36 -> Is a Prime Factor, and highest (Correct Answer)

© 252/5 = fraction -> Not a prime factor

(B) 252/3 = fraction -> Not a prime factor

(A) 252/2 = 126 -> Is a Prime Factor, but not the highest

 

Oddly enough this works if you solve it for the base of 2 instead lets consider.

 

Step 1:Create equal -> 4^17 - 2^28 is equivalent to

(2^17)(2^17) - (2^28) which is also equal to (2^34) - (2^28)

Step 2: Subtract the exponents; 34-28 = 6

Step 3: Next, take an easier example to get to a 6 exponent difference still using the number 2 as your base.

Example: 2^7 - 2^1 = 128 - 2 = 126

Step 4: Now, working backwards from the answer choices, starting with (E), divide 126 by all answer choices and see what is the highest prime factor to divide into 126.

(E) 126/11 = fraction -> Not a prime factor

(D) 126/7 = 36 -> Is a Prime Factor, and highest (Correct Answer)

© 126/5 = fraction -> Not a prime factor

(B) 126/3 = 42 -> Is a Prime Factor, but not the highest

(A) 126/2 = 126 -> Is a Prime Factor, but not the highest

 

Seems to work with all examples. Am I crazy, or is this really a shortcut...inquiring minds (me) want to know!!!!!

 

 

[hitchhiker3000]

I don't know why you have to do all these to arrive at the answer.

 

4^17 - 2^28 = 4^14(4^3 - 1)

= 4^14(63)

The biggest prime no. in the given list that will be a factor of this will be 7.

as 63 = 7*9

 

In a way what you have done is similar to this but involves more calculation.

The difference between the exponents is 3 here also (4^3 - 4^0)

[jordan23VT]

excellent approach hitchiker

[0riginal]

What is the greatest prime factor of (4^17) - (2^28)?

 

A) 2

B) 3

C) 5

D) 7

E) 11

 

4^17-2^28 = 2^34-2^28 = 2^28 (2^6 - 1) = 2^28 * 63 = 2^28 *7*3*3

 

Hence, 7