What is the greatest prime factor of 6^8−3^8 ?

Question

What is the greatest prime factor of  6^8−3^8  ?

A) 3

B) 11

C) 17

D) 19

E) 31

zxcvbnmas · Accepted Answer

6^8 - 3^8

2^8*3^8 - 3^8

3^8(2^8 - 1)

(2^4 + 1)(2^4 - 1)

(16+1)(16-1)

(17)(15)

So 17 is Answer

BrentGMATPrepNow · Answer

6⁸ − 3⁸ is a DIFFERENCE OF SQUARES. So we can factor it. 
6⁸ − 3⁸ = (6⁴ + 3⁴)(6⁴ - 3⁴)
= (6⁴ + 3⁴)(6² + 3²)(6² - 3²)
= (6⁴ + 3⁴)(6² + 3²)(6 + 3)(6 - 3)
= (6⁴ + 3⁴)(45)(9)(3)
= ( 6⁴ + 3⁴ )(3)(3)(5)(3)(3)(3)

Hmmmm, we can see that the correct answer is "hiding" in the first number ( 6⁴ + 3⁴ )
Let's factor out the 3⁴, to get:
 6⁴ + 3⁴ = 3⁴(2⁴ + 1)
= 3⁴(16 + 1)
= 3⁴(17)
= (3)(3)(3)(3)(17)

So, 6⁸ − 3⁸ =  (3)(3)(3)(3)(17) (3)(3)(5)(3)(3)(3)

So the correct answer is C

Cheers,
Brent

AdlaT · Answer

Solution:

6^8-3^8=3^8*(2^8-1)=3^8(256-1)=3^8(255)=3^8*51*5=3^8*17*3*5 .

ANS .17

jonmarrow · Answer

how do u solve this? how do u know what methodology to apply when u get a question like this?
im soo confused!!

Bunuel · Answer

To find the greatest prime of 6^8−3^8 you should make prime factorization of this number. The techniques above does exactly this. Check similar questions to practice: what-is-the-greatest-prime-factor-of-158991.html what-is-the-greatest-prime-factor-of-104757.html what-is-the-greatest-prime-factor-of-70126.html what-is-the-greatest-prime-factor-of-190425.html Hope it helps.

LogicGuru1 · Answer

First figure out the common factors in both 
 6^8  can be written s  2^8 * 3^8

take 3^8 common

3^8*2^8-3^8 
 3^8(2^8-1)  (here you should know that  2^8 = 256 , The easiest way to remember this is  2^{10}=1024  and now you can derive most  2^x )
SO your expression becomes  3^8(256-1)  ===>  3^8 (255) ==> 3^8 (17*15) 
so your prime factorisation will be  3^8*15*17 ===> 3^8*(3^1*5^1)*17 =====> 3^9*5^1*17

so as we can see there are there prime numbers here 3, 5 and 17 out of which 3 is the smallest prime number and 17 is the biggest.

Hence 17 is the biggest Prime Factor

Answer is C

CounterSniper · Answer

3^8(2^8-1)=3^8(255) =3^8(17*15)=3^8*17*5*3=3^9*5*17 
therefore 17 C

ramanan42 · Answer

what about 3^8, it could have a prime factor greater than 17.

ramanan42 · Answer

understood the reasoning behind this . ! good now!

Nunuboy1994 · Answer

It's interesting to see all the different methodologies here- many paths that lead to the same answers; anyways, my method following this technique was slightly different. When you mention the answer lies in (6^4 + 3^4)

3^4(3^4 +1)
81 (82)
82/2 =
41/3 = 17 (plug in values)

ydmuley · Answer

In order to find the greatest prime factor, lets break down the below number into its prime factors

6^8−3^8

=  (3^8 * 2^8) − 3^8

=  3^8 (2^8 - 1)

=  3^8 * 255

=  3^8 * 15 * 17

=  3^9 * 5 * 17

As we can see that the greatest prime factor is  17

Answer is C

EMPOWERgmatRichC · Answer

Hi All,

If you're comfortable with Exponent Rules and Factoring, you can also approach this prompt without using a Quadratic:

6^8 - 3^8 =

First, we can 'rewrite' 6^8.... 
(2^8)(3^8) - 3^8

Now, factor out 3^8... 
(3^8)(2^8 - 1)

2^8 = 256 
(3^8)(256-1) 
(3^8)(255) 
(3^8)(5)(51) 
(3^8)(5)(3)(17)

The greatest prime factor is 17.

Final Answer:  C

GMAT assassins aren't born, they're made, 
Rich

nigina93 · Answer

6^8-3^8=
3^8*2^8-3^8=
3^8(2^8-1)=
3^8(256-1)=
3^8(255)=
3^8(3*5*17)
Answ is 17 (C)

Abhishek009 · Answer

6^8−3^8

=  2^8*3^8−3^8

=  3^8(2^8 −1)

Now,  2^8 − 1   = 255 = 15*17 , So The greatest Prime number is 17 Answer must be (C)

London1 · Answer

It should be 17, since we can use a^2 - b^2 here. C

LeoN88 · Answer

Prime factor is a number that is divisible by only one and itself; e.g. 3, 5, 19 etc.

We have to break down this number as far as possible.

3^8 * (2^8 - 1)
3^8 * (255)
3^8 * (51*5)
3^8 * (5)(17*3)

So, this number can be factorized into following primes: 3, 5, 17. 17 being the greatest.

ScottTargetTestPrep · Answer

Solution:
 
Let’s prime factorize the given expression:
 
6^8 - 3^8

3^8(2^8 - 1)

3^8(2^4 - 1)(2^4 + 1)

3^8(16 - 1)(16 + 1)

3^8(15)(17)

3^8 x 3 x 5 x 17

So the largest prime factor is 17.

Answer: C

Basshead · Answer

6^8 - 3^8

=  (3^8 * 2^8) - 3^8

3^8 ( 2^8 - 1)

3^8 (2^4 +1)(2^4-1)

3^8 * 17 * 15

Greatest prime factor is 17. Answer is C.

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What is the greatest prime factor of 6^8−3^8 ?

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