using fermat's little theorum and problem solving

Mar 2010
8
0
Give two different methods for finding 2^999(mod 5), one using Fermat's little theorem and the other using basic problem solving methods.
 
Nov 2009
485
184
I'll give you a hint for a basic method. Write \(\displaystyle 2^{999}\equiv 2\cdot 2^{998}\equiv 2\cdot 4^{499}\pmod{5}\). Can you compute this directly?
 
Jul 2009
69
6
Give two different methods for finding 2^999(mod 5), one using Fermat's little theorem and the other using basic problem solving methods.
Using Fermats little theorem \(\displaystyle 2^4 \equiv 1 \pmod{5} \)
Then \(\displaystyle (2^4)^{249} \equiv 1 \pmod{5} \)
Finally \(\displaystyle (2^4)^{249}*2^3 \equiv 1*2^3 \pmod{5} \)
therefore \(\displaystyle 2^3 \equiv 3 \pmod{5} \)
So \(\displaystyle 2^{999} \equiv 3 \pmod{5} \)