Modern Algebra question

Printable View

October 27th 2009, 05:19 PM
jujab

Modern Algebra question

Evaluate modulo 5.

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

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

Could someone help me please

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?
October 27th 2009, 08:07 PM
tonio

Quote:

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]}.

Could someone help me please

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:

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

Tonio
October 28th 2009, 10:30 AM
jujab

So then it would be 3^4*3 => 3^5 => 3mod5 right?
October 28th 2009, 11:03 AM
tonio

Quote:

Originally Posted by jujab

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

Yup

Tonio
October 28th 2009, 11:18 AM
jujab

Thank you Tonio!
October 28th 2009, 03:39 PM
Bingk

Or you could use Fermat's little theorem :)

$3^{101} \equiv 3^{100} \cdot 3 \equiv (3^{25})^4 \cdot 3 \equiv 1 \cdot 3 \equiv 3 \ (mod \ 5)$
October 28th 2009, 04:45 PM
jujab

Thanks Bingk! Much cleaner!!