Chinese Remainder Theorem
Mathematica, 55 51 45
Modular inverse is banned, but modular exponentiation is allowed. By Fermat's little theorem, n^(-1) % p == n^(p-2) % p.
(PowerMod[x=1##&amp;@@#/#,#-2,#]x).#2&amp;@@Thread@#&amp;

Example:
In[1]:= f = (PowerMod[x=1##&amp;@@#/#,#-2,#]x).#2&amp;@@Thread@#&amp;;

In[2]:= f[{{5, 3}}]

Out[2]= 3

In[3]:= f[{{7, 2}, {5, 4}, {11, 0}}]

Out[3]= 1584

In[4]:= f[{{5, 1}, {73, 4}, {59, 30}, {701, 53}, {139, 112}}]

Out[4]= 142360350966


Just for fun:
ChineseRemainder@@Reverse@Thread@#&amp;