GMAT Prep 2 - Prime factor

4 ^ 17 - 2 ^18
= 4^17 - 4^9
= 4^9 (4^8 -1)

We need to find the greatest prime factor of 4^8-1
4^8-1 = 4^8-1^2

Lets look at 4^8

4^8 = 16^4 = 256^2

hence, 4^8-1^2 = 256^2 - 1^2
= (256-1) (256+1)
= 255 x 257

now, just factor 255 and 257

= (5 x 3 x 17) x 257 (257 is a prime since it has no factors from 1 - 17)

so the answer should be 257

I think you got the question wrong. The correct question is:

What is the Greatest Prime Factor of 4^17 - 2^28?

you can solve the problem with the same approach

Yes it seems there is some problem with the ques.

The ans should be 257.

netigen wrote:4 ^ 17 - 2 ^18
= 4^17 - 4^9
= 4^9 (4^8 -1)

You seem quite used to juggling with exponents and square roots...There are rules that I'm not aware of. Could you advice me an online reference to take a loot at where I could learn all these algebraic rules please?

I just ordered the GMAT 800 book, do you think there is a list of all these advanced rules that one can use in Math?

Thanks :roll:

Yes most of the GMAT books have these rules. In case you have difficulty in understanding these steps

4 ^ 17 - 2 ^18
= 4^17 - 4^9
= 4^9 (4^8 -1)

I would suggest to start with the Maths workbook from Kaplan rather than the GMAT 800 that you have ordered.

Thanks! Will do

4 ^ 17 - 2 ^18
= 4^17 - 4^9
= 4^9 (4^8 -1)

Hey! netigen.. could u please explain how did u leave 4^9 and start finding the greatest factor of 4^8-1?

We need to find the greatest prime factor of 4^8-1
4^8-1 = 4^8-1^2

and again here how did u come up with 1^2 ? please explain
Lets look at 4^8

4^8 = 16^4 = 256^2

hence, 4^8-1^2 = 256^2 - 1^2
= (256-1) (256+1)
= 255 x 257

now, just factor 255 and 257

= (5 x 3 x 17) x 257 (257 is a prime since it has no factors from 1 - 17)

so the answer should be 257

I was trying to reduce the eq to the form a^2 - b^2
1^2 = 1 (thats why you can substitute either for the other)

since 2 is the only prime factor of 4^9 we know that 2 is the factor of the final number. So I left that out and looked at the other factors to find out if there is a prime factor greater than 2 available.

Let me know if this makes sense.