Question

Someone please tell me what is Euler's Theorem?

Answer 1

Hey there!

→ EULER'S THEOREM:
Let a and m be coprime. Then aφ(m) = 1 (mod m).

→ PROOF:
The proof is completely analogous to that of the Fermat's Theorem except that instead of the set of non-negative remainders {1,2,...,m-1} we now consider the set {m1, m2, ..., mφ(m)} of the remainders of division by m coprime to m. In exactly the same manner as before, multiplication by a results in a permutation (but now) of the set{m1, m2, ..., mφ(m)}. Therefore, two products are congruent:

m1m2 ... mφ(m) = (am1)(am2) ... (amφ(m)) (mod m)

dividing by the left-hand side proves the theorem.

Hope it helped
#EshanSingh1

Answer 2

For understanding Euler theorem ,first try to understand what is Euler number ?
Let a number P ,
Euler number of P , is always then P, and co-prime to P .
example :
6 have two Euler number 3, 5 because all are less then 6 and co-prime .

finding Euler number ?
use formula ,
E(P) = P×( 1 - 1/x )( 1 - 1/y)( 1 - 1/z).....
where P is the a number ,
x , y z ,..are prime factor of P .

now,
Euler theorem :- this is great theorem . we can easily use for answering remainder questions .

" when Qe(P) is divided by P, then remainder always will be 1. "
where e(P) is Euler number of P,
P and Q are co-prime to each other .
___________________________________________________
example :-
find the remainder of 13²² when divided by 23 ?
here 22 is Euler number 23
[ euler number of prime number is alwys less then 1 by that number
e(P) = 23 - 1 = 22
also 13, and 23 are co- prime to each other.

so by Euler theorem remainder = 1