# Math Help - using fermat's little theorum and problem solving

1. ## using fermat's little theorum and problem solving

Give two different methods for finding 2^999(mod 5), one using Fermat's little theorem and the other using basic problem solving methods.

2. I'll give you a hint for a basic method. Write $2^{999}\equiv 2\cdot 2^{998}\equiv 2\cdot 4^{499}\pmod{5}$. Can you compute this directly?

3. Originally Posted by apple2010
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 $2^4 \equiv 1 \pmod{5}$
Then $(2^4)^{249} \equiv 1 \pmod{5}$
Finally $(2^4)^{249}*2^3 \equiv 1*2^3 \pmod{5}$
therefore $2^3 \equiv 3 \pmod{5}$
So $2^{999} \equiv 3 \pmod{5}$