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Difficulty:
35%
(medium)
Question Stats:
67% (01:37) correct
33%
(01:43)
wrong
based on 1479
sessions
History
Date
Time
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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
A) 3
B) 11
C) 17
D) 19
E) 31
\(6^8 - 3^8\)
\(2^8*3^8 - 3^8\)
\(3^8(2^8 - 1)\)
\((2^4 + 1)(2^4 - 1)\)
\((16+1)(16-1)\)
\((17)(15)\)
So 17 is Answer
\(2^8*3^8 - 3^8\)
\(3^8(2^8 - 1)\)
\((2^4 + 1)(2^4 - 1)\)
\((16+1)(16-1)\)
\((17)(15)\)
So 17 is Answer
Posts: 6,727
Updated on: 18 Jun 2021, 05:13
Originally posted by BrentGMATPrepNow on 15 Feb 2016, 12:55.
Last edited by BrentGMATPrepNow on 18 Jun 2021, 05:13, edited 1 time in total.
Last edited by BrentGMATPrepNow on 18 Jun 2021, 05:13, edited 1 time in total.
Quote:
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
General Discussion
zxcvbnmas
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
how do u solve this? how do u know what methodology to apply when u get a question like this?
im soo confused!!
im soo confused!!
jonmarrow
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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Posts: 463
Updated on: 05 Apr 2018, 23:20
Originally posted by LogicGuru1 on 10 Jul 2016, 00:40.
Last edited by LogicGuru1 on 05 Apr 2018, 23:20, edited 1 time in total.
Last edited by LogicGuru1 on 05 Apr 2018, 23:20, edited 1 time in total.
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
\(6^8\) 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
zxcvbnmas
Posts: 609
3^8(2^8-1)=3^8(255) =3^8(17*15)=3^8*17*5*3=3^9*5*17
therefore 17 C
therefore 17 C
AdlaT
what about 3^8, it could have a prime factor greater than 17.
ramanan42
understood the reasoning behind this . ! good now!
GMATPrepNow
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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zxcvbnmas
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
EMPOWERgmatRichC
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Expert reply
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:
GMAT assassins aren't born, they're made,
Rich
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:
GMAT assassins aren't born, they're made,
Rich
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)
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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zxcvbnmas
\(6^8−3^8\)
= \(2^8*3^8−3^8\)
= \(3^8(2^8 −1)\)
Now, \(2^8 − 1\) \(= 255 = 15*17\), So The greatest Prime number is 17 Answer must be (C)
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It should be 17, since we can use a^2 - b^2 here. C
zxcvbnmas
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
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Expert reply
Posts: 22,739
zxcvbnmas
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
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zxcvbnmas
\(6^8 - 3^8\)
= \((3^8 * 2^8) - 3^8\)
\(3^8 ( 2^8 - 1)\)
\(3^8 (2^4 +1)(2^4-1)\)
\(3^8 * 17 * 15\)
Greatest prime factor is 17. Answer is C.
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