1. ## Lagrange's theorem

Z6={1,2,3,4,5,} (modulo 6) the action * - is multiply. n=|Z|
By Lagrange's theorem - for every number that belong to Z6 (lets say x). so x^n sepopse to be the neutral number. so x^n sepose to be 1.
but 5^5=3125 -> 3125/6=3120 -> 3125-3120=5 -> 5^5 (modulo 6) ≠ 1
same with the other numbers....

so what am i doing wrong???

2. Originally Posted by tukilala
Z6={1,2,3,4,5,} (modulo 6) the action * - is multiply. n=|Z|
By Lagrange's theorem - for every number that belong to Z6 (lets say x). so x^n sepopse to be the neutral number. so x^n sepose to be 1.
but 5^5=3125 -> 3125/6=3120 -> 3125-3120=5 -> 5^5 (modulo 6) ≠ 1
same with the other numbers....

so what am i doing wrong???
$Z_6 = \{0,1,2,3,4,5\}$

order is $6$ ... $5^6 = 15625 \equiv 1(mod \, 6)$

3. ok
so now it's have 5 elements
Z6/{0} = {1,2,3,4,5}
order is 5... 5^5 (modulo 6) ≠ 1

4. $\mathbb{Z}_6^\times$ is not a group under multiplication. There are no inverses to 2 and 3.

$\mathbb{Z}_n^\times$ is a group if and only if n is prime. In that case it has order p-1 and indeed you can check that $a^{p-1}\equiv 1 \mod p$ for all $a$ not divisible by p. This is what Fermat's "little theorem" says; it's actually a very specific case of Lagrange's theorem.

5. lets say that i need to find the last 3 digit of the number: 998^1010
i know that to find the last 3 digits, i need to use modulo 1000
so im looking at the group Z1000 under multiplication, right?
and now: 998^1000 = 998^(999+11) = (998^999)*(998^11)
is it true to say that 998^999 = 1 (by Lagrange's theorem)?? if yes, so why in the group Z6/{0} = {1,2,3,4,5} under multiplication 5^5≠1 (mod 5). 1000 is not a prime number, so why in this case Lagrange's throrem works?

any way, if its true to say that( and i dont understant yet why) so 998^11 = -2^11 = 1024*(-2) = -48 = 952 (mod 1000)
so the last 3 numbers are:952

so when i can use Lagrange's theorem and when not?

6. Originally Posted by tukilala
lets say that i need to find the last 3 digit of the number: 998^1010
i know that to find the last 3 digits, i need to use modulo 1000
so im looking at the group Z1000 under multiplication, right?
and now: 998^1000 = 998^(999+11) = (998^999)*(998^11)
is it true to say that 998^999 = 1 (by Lagrange's theorem)?? if yes, so why in the group Z6/{0} = {1,2,3,4,5} under multiplication 5^5≠1 (mod 5). 1000 is not a prime number, so why in this case Lagrange's throrem works?

any way, if its true to say that( and i dont understant yet why) so 998^11 = -2^11 = 1024*(-2) = -48 = 952 (mod 1000)
so the last 3 numbers are:952

so when i can use Lagrange's theorem and when not?
Did you read my post? That is not a group.

7. Originally Posted by tukilala
998^1000 = 998^(999+11) = (998^999)*(998^11)
is it true to say that 998^999 = 1 (by Lagrange's theorem)??
$998^{999} \equiv 312 mod 1000$...as Bruno said, using multiplication modulo $n$, Lagrange's theorem holds if and only if $n$ is prime. In this case we call it "Fermat's Little Theorem".

I always double check my working in such problems with a calculator. Here the numbers are big so I used Maple, but you could try either Wolfram Alpha or the calculator function on google (although I couldn't get google to work for this example).

8. Wolfam Alpha is nice..
but when im doing a test, i dont have this web site to calculate for me those numbers...
so how can i calculate
998^999 (mod 1000)
without internet
??

9. Originally Posted by tukilala
but when im doing a test, i dont have this web site to calculate for me those numbers...
No, but the "trick" is to get so well versed in such questions as to, eventually, not need such a program.

Originally Posted by tukilala
so how can i calculate
998^999 (mod 1000)
without internet
??
I think the answer to this is: not easily.

Note that $1010$ is even and so $2^{1010} = (-2)^{1010}$.

Hint: prove that $2^3(2^{100}-1) \equiv 0 mod 1000$ (note that $24=3*8$).

The problem with this solution is that this hint was not easy to either "see" or prove without a calculator.

There may be a more elegant answer, but that's the best I can conjure up at the moment.