Some facts

1) For any number N=Sigma(p_i), i=1..k, p_i are prime factors of N with any prime number cp, cp is not factor of N, there is:
Mod(cp^LCM(p_i-1, i=1..k), N)=1.
for some case, Mod(cp^(LCM(p_i-1, i=1..k)/2), N)= +/-1

Special: when N is prime, Mod(cp^(N-1), N)=1

For example:
In[3]:= FactorInteger[66855224152]
Out[3]= {{2, 3}, {19, 1}, {1549, 1}, {283949, 1}}
In[5]:= LCM[18,1548,283948]
Out[5]= 109887876
In[6]:= Mod[3^109887876,66855224152]
Out[6]= 1
In[8]:= Mod[5^109887876,66855224152]
Out[8]= 1
In[9]:= Mod[5^(109887876/2),66855224152]
Out[9]= 1
In[10]:= Mod[7^109887876,66855224152]
Out[10]= 1