1. Modern Algebra question

Evaluate modulo 5.

[(4+6)9 + 12(124)]^101

write as an element of the set {[0],[1],[2],[3],[4]}.

Would it be 1578 ^ 1
which would be 3 modulo 5? or am I way off? Also, how would I write it as an element of the set above?

2. Originally Posted by jujab
Evaluate modulo 5.

[(4+6)9 + 12(124)]^101

write as an element of the set {[0],[1],[2],[3],[4]}.

Would it be 1578 ^ 1
which would be 3 modulo 5? or am I way off? Also, how would I write it as an element of the set above?

Doing arithmetic modulo 5, we get that:

$\displaystyle [(4+6)9 + 12(124)]^{101} = [(4+1)\cdot 4 + 2\cdot 4)^{101} $$\displaystyle = 3^{101} = (3^{20})^5\cdot 3=$$\displaystyle 3^{20}\cdot 3=(3^5)^4\cdot 3=3^4\cdot 3=...$ ...and now you end the argument.

Tonio

3. So then it would be 3^4*3 => 3^5 => 3mod5 right?

4. Originally Posted by jujab
So then it would be 3^4*3 => 3^5 => 3mod5 right?

Yup

Tonio

5. Thank you Tonio!

6. Or you could use Fermat's little theorem

$\displaystyle 3^{101} \equiv 3^{100} \cdot 3 \equiv (3^{25})^4 \cdot 3 \equiv 1 \cdot 3 \equiv 3 \ (mod \ 5)$

7. Thanks Bingk! Much cleaner!!