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

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What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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

A) 3

B) 11

C) 17

D) 19

E) 31
[Reveal] Spoiler: OA
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Re: What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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New post 15 Feb 2016, 12:55
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Quote:

What is the greatest prime factor of 6⁸ − 3⁸?

a) 3
b) 11
c) 17
d) 19
e) 31

Thanks in advance :-)


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

ASIDE: For more on factoring differences of squares, see our free video - http://www.gmatprepnow.com/module/gmat- ... /video/955

Cheers,
Brent
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What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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New post 15 Feb 2016, 12:56
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\(6^8 - 3^8\)

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

\(3^8(2^8 - 1)\)

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

\((16+1)(16-1)\)

\((17)(15)\)

So 17 is Answer

Last edited by zxcvbnmas on 15 Feb 2016, 13:03, edited 1 time in total.
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Re: What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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New post 15 Feb 2016, 13:01
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zxcvbnmas wrote:
What is the greatest prime factor of \(6^8−3^8\) ?

A) 3

B) 11

C) 17

D) 19

E) 31


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
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Re: What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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New post 09 Jul 2016, 12:05
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how do u solve this? how do u know what methodology to apply when u get a question like this?
im soo confused!!
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What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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New post 09 Jul 2016, 22:27
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jonmarrow wrote:
how do u solve this? how do u know what methodology to apply when u get a question like this?
im soo confused!!


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.
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Re: What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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New post 10 Jul 2016, 00:40
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 \(3^8*15*17\)===>\(3^9*5^1*17\)

Hence 17 is the biggest Prime Factor

Answer is C


zxcvbnmas wrote:
What is the greatest prime factor of \(6^8−3^8\) ?

A) 3

B) 11

C) 17

D) 19

E) 31

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Re: What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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3^8(2^8-1)=3^8(255) =3^8(17*15)=3^8*17*5*3=3^9*5*17
therefore 17 C
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Re: What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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New post 14 Jul 2016, 19:57
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AdlaT wrote:
zxcvbnmas wrote:
What is the greatest prime factor of \(6^8−3^8\) ?

A) 3

B) 11

C) 17

D) 19

E) 31


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



what about 3^8, it could have a prime factor greater than 17.
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Re: What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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New post 14 Jul 2016, 20:08
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ramanan42 wrote:
AdlaT wrote:
zxcvbnmas wrote:
What is the greatest prime factor of \(6^8−3^8\) ?

A) 3

B) 11

C) 17

D) 19

E) 31


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



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


understood the reasoning behind this . ! good now!
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Re: What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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New post 10 Apr 2017, 17:49
GMATPrepNow wrote:
Quote:

What is the greatest prime factor of 6⁸ − 3⁸?

a) 3
b) 11
c) 17
d) 19
e) 31

Thanks in advance :-)


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

ASIDE: For more on factoring differences of squares, see our free video - http://www.gmatprepnow.com/module/gmat- ... /video/955

Cheers,
Brent


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)
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Re: What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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zxcvbnmas wrote:
What is the greatest prime factor of \(6^8−3^8\) ?

A) 3

B) 11

C) 17

D) 19

E) 31


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

\(6^8−3^8\)

= \((3^8 * 2^8) − 3^8\)

= \((3^8 * 2^8) − 3^8\)

= \(3^8 (2^8 - 1)\)

= \(3^8 * 255\)

= \(3^8 * 15 * 17\)

= \(3^9 * 5 * 17\)

As we can see that the greatest prime factor is \(17\)

Answer is C
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Re: What is the greatest prime factor of 6^8−3^8 ? [#permalink]

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New post 08 Feb 2018, 11:38
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:
[Reveal] Spoiler:
C


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Re: What is the greatest prime factor of 6^8−3^8 ?   [#permalink] 08 Feb 2018, 11:38
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