EULER'S TOTIENT FUNCTION AND CONGRUENCE THEOREM 
                            GENERALIZED BY SMARANDACHE



          Let a, m be integers, m different from 0.  Then:

     phi(m ) + s
          s                        s
    a             is congruent to a   (mod m),

    where phi(x) is Euler's Totient Function,

    and s and m  are obtained through the algorithm:
               s

       ___
      |    a = a d      ;  (a , m ) = 1;
      |         0 0          0   0
  (0) |
      |    m = m d      ;   d  different from 1;
      |         0 0          0
       ---

       ___       1            1
      |    d  = d d      ;  (d , m ) = 1;
      |     0    0 1          0   1
  (1) |
      |    m  = m d      ;   d  different from 1;
      |     0    1 1          1
       ---

  ....................................................

       ___         1              1
      |    d    = d   d      ;  (d   , m   ) = 1;
      |     s-2    s-2 s-1        s-2   s-1
 (s-1)|
      |    m    = m   d       ;   d    different from 1;
      |     s-2    s-1 s-1         s-1
       ---

       ___         1              1
      |    d    = d   d      ;  (d   , m ) = 1;
      |     s-1    s-1 s          s-1   s
 (s)  |
      |    m    = m d       ;   d  = 1.
      |     s-1    s s           s
       ---

Therefore it is not necessary for a and m to be coprime.

            25604
Example:   6      is congruent to ? (mod 105765).
           It is not possible to use Fermat's or Euler's theorems,
           but the Smarandache Congruence Theorem works:

           d  = (6, 105765) = 3   
            0
           m  = 105765/3 = 35255
            0
           i = 0
           3 is different from 1


           thus i = 0+1 = 1
           d  = (3, 35255) = 1
            1
           m  = 35255/1 = 35255.
            1

                      phi(35255)+1                  1
           Therefore 6             is congruent to 6  (mod 105765)

                   25604               4
           whence 6      is congruent 6  (mod 105765).


References:

 [1] Porubsky, Stefan, "On Smarandache's Form of the Individual Fermat-
      Euler Theorem", <Smarandache Notions Journal>, Vol. 8, No. 1-2-3,
      Fall 1997, pp. 5-20, ISSN 1084-2810.
 [2] Porubsky, Stefan, "On Smarandache's Form of the Individual Fermat-
      Euler Theorem", <Proceedings of the First International Conference
      on Smarandache type Notions in Number Theory>, University of Craiova,
      August 21-24, 1997, pp. 163-178, ISBN 1-879585-58-8.
 [3] Smarandache, Florentin, "Une generalization de theoreme d'Euler"
      (French), <Buletinul Universitatii Brasov>, Seria C, Vol. XXIII,
      1981, pp. 7-12, reviewed in Mathematical Reviews: 84j:10006.
 [4] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus,
      Bucharest, 1996, pp. 182-191.