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Manager  Joined: 06 Jun 2014
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What is the greatest prime factor of 6^8−3^8 ?  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 63% (01:39) correct 37% (01:38) wrong based on 502 sessions

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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
GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4015
Re: What is the greatest prime factor of 6^8−3^8 ?  [#permalink]

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9
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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19
7
$$6^8 - 3^8$$

$$2^8*3^8 - 3^8$$

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

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

$$(16+1)(16-1)$$

$$(17)(15)$$

So 17 is Answer

Originally posted by zxcvbnmas on 15 Feb 2016, 13:56.
Last edited by zxcvbnmas on 15 Feb 2016, 14:03, 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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1
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
Intern  Joined: 22 Jun 2016
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Re: What is the greatest prime factor of 6^8−3^8 ?  [#permalink]

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1
how do u solve this? how do u know what methodology to apply when u get a question like this?
im soo confused!!
Math Expert V
Joined: 02 Sep 2009
Posts: 58434
What is the greatest prime factor of 6^8−3^8 ?  [#permalink]

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5
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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GMAT 1: 750 Q49 V43 What is the greatest prime factor of 6^8−3^8 ?  [#permalink]

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1
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

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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Originally posted by LogicGuru1 on 10 Jul 2016, 01:40.
Last edited by LogicGuru1 on 06 Apr 2018, 00:20, 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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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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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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Posts: 6
Re: What is the greatest prime factor of 6^8−3^8 ?  [#permalink]

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ramanan42 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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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 $$17$$

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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: What is the greatest prime factor of 6^8−3^8 ?  [#permalink]

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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.

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

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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)
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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

$$6^8−3^8$$

= $$2^8*3^8−3^8$$

= $$3^8(2^8 −1)$$

Now, $$2^8 − 1$$ $$= 255 = 15*17$$, So The greatest Prime number is 17 Answer must be (C)
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Re: What is the greatest prime factor of 6^8−3^8 ?  [#permalink]

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

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.
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