Problem Solving

What is the greatest prime factor of 4^17 - 2^26?

a) 2
b) 3
c) 5
d) 7
e) 11

Source: GMAT Prep test

Thanks,
Vito

Probably you should try using this

2 ^ 2 ^ 17 - 2 ^ 26 = 2 ^ 26 (2 ^ 8 - 1) = 2 ^ 26 x 255

rajs.kumar wrote:Probably you should try using this

2 ^ 2 ^ 17 - 2 ^ 26 = 2 ^ 26 (2 ^ 8 - 1) = 2 ^ 26 x 255

I don't see how that would solve the problem above. Please advise.

4^17 - 2 ^ 26

(or) 2^34 - 2 ^26

(or) 2^26(2^8-1)

(or) 2*2^25*255

Factors of 255 are 1,3,5,15,17,51,85,255

Hence the prime factors of the whole expression are 2,3,5,17.

Highest is 5 (among the answer choices provided)

Therefore, should be C (as per my understanding of this problem).

first, you need to understand what prime factors are so as to enhance your understanding on this question and subsequent questions later that you may lay your hands on.
prime numbers which are divisible only by itself and 1. examples are 2,3,5,7,11,13,19,23 and so on.

for this question 4^17 - 2^26
the figure is not a prime number. so it must be reduce to its prime factors. prime factors deal with the finding of a prime number we will need to multiply to get the original number.
prime factors of 4= 2*2 =2^2
4^17 - 2^26
2^(2*17) - 2^26
2^34 - 2^26
2^26(2^8-1)
2^26* (256-1)
2^26* (255)
Now to break 255 to its simplest from, we need to find what its prime factors are
255=1*3*5*15*17*51*85*255
therefore,
2^26 * (3*85)
2^26 *(3*5*17)
working directly with the options given,
prime factor 5 is the greatest prime factor.

so, option C is the correct

For this exercise we need to know this property of the powers:
((a)^n)^m= ((a)^m)^n=(a)^n*m.

Also, there is a formula that can help us to solve this exercise, which is:
a^2-b^2=(a-b)*(a+b).

We have this expression: 4^17-2^26.

Now we can write: 4=2^2 and using the red formula at the beginning we will get:
4^17-2^26= ((2)^2)^17-2^26 = 2^(2*17)-2^26 = 2^34 - 2^26= 2^26(2^8-1).

Now, 2^8-1 = (2^4)^2 - 1^2, and using the green formula we will get that
2^8-1 = (2^4)^2 - 1^2 = (2^4 - 1)*(2^4 + 1) = (16-1)*(16+1) = 15*17.

Replacing the last line in the black expression we will get:
4^17-2^26 = 2^26(2^8-1) = (2^26)*(15)*17.

Now, we need to write this last expression as a product of prime numbers. This is,
(2^26)*(3)*(5)*(17).

The prime factor of 2^26 is only 2. So the prime factors of the orange expression are 2, 3, 5 and 17. So, the greatest prime factor is 17, but according to the given options, the answer is C.