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Problem solving - Primenumbers

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by lucas211 » Sun Feb 07, 2016 6:10 am
Hello BTG

Would appreciate a little help on the following question:

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

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

Thanks in advance :-)
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by Brent@GMATPrepNow » Sun Feb 07, 2016 8:56 am
lucas211 wrote:Hello BTG

Would appreciate a little help on the following question:

What is the greatest prime factor of 6� − 3�?

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

Thanks in advance :-)
6� − 3� is a DIFFERENCE OF SQUARES. So we can factor it.
6� − 3� = (6� + 3�)(6� - 3�)
= (6� + 3�)(6² + 3²)(6² - 3²)
= (6� + 3�)(6² + 3²)(6 + 3)(6 - 3)
= (6� + 3�)(45)(9)(3)
= (6� + 3�)(3)(3)(5)(3)(3)(3)

Hmmmm, we can see that the correct answer is "hiding" in the first number (6� + 3�)
Let's factor out the 3�, to get:
6� + 3� = 3�(2� + 1)
= 3�(16 + 1)
= 3�(17)
= (3)(3)(3)(3)(17)

So, 6� − 3� = (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 - https://www.gmatprepnow.com/module/gmat- ... /video/955

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by [email protected] » Sun Feb 07, 2016 9:57 am
Hi lucas211,

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

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by lucas211 » Tue Feb 09, 2016 4:14 am
Thanks to both of you. Really appreciate it.
This is definitely one of my soft spot.

@Rich

Why do we not try to break down the number 3^8?

Thanks again
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by [email protected] » Tue Feb 09, 2016 8:56 am
Hi lucas211,

3^8 is just a 'bunch' of 3s, so there's no real work to be done there:

3^8 = (3)(3)(3)(3)(3)(3)(3)(3)

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