Greatest Prime Factor of 4^17 - 2^28

[Mixedlatina20]

Hello,

A friend of mine has this problem...

What is the Greatest Prime Factor?

4^17 - 2^28

He knows that the answer is 7 but doesn't know why or what to do to get that answer. I tried myself but I haven't done this kind of math in a long while. Can you please help? Thanks

 

[galactus]

Re: Greatest Prime Factor

\(\displaystyle 4^{17} - 2^{28} = (2^{2})^{17} - 2^{28}\)

\(\displaystyle = 2^{34} - 2^{28}\)

\(\displaystyle = 2^{28}\cdot 2^{6} - 2^{28}\)

\(\displaystyle = 2^{28}\cdot (2^{6} - 1)\)

\(\displaystyle = 2^{28}\cdot 63\)

\(\displaystyle = 2^{28}\cdot 3^{2}\cdot 7\)

Therefore, we can see 7 is the greatest prime factor...9 and 2^28 are not prime.

 

[Mixedlatina20]

Re: Greatest Prime Factor

Thank you so much for the replay but can you tell me how you went from

2^28 ? 2^6 - 2^28 to 2^28 ? (2^6 - 1)? How did u get the 1? After that point I understand but am lost with how to get 1?

 

[galactus]

Re: Greatest Prime Factor

I factored out the 2^28.

 

[Denis]

Re: Greatest Prime Factor

As simple as: a^2 - a = a(a - 1) ; kapish?