Here are some of the things you should review for the midterm: 

Euclidean Algorithm, how to find greatest 
common divisor g = (m,a) and integers x,y such that g=mx+ay. 

Primes, Prime factorization, Primes in arithmetic progression, 
Fermat's little theorem and Euler's generalization, Wilson's Theorem 

Pseudoprimality Test

Euler Phi-Function and its properties

Congruences, 
Chinese remainder theorem, 
Computational Techniques, 
Public-Key Cryptography algorithm. 

Solving congruences, explicitely solve ax=b (mod m)
Quadratic congruences mod primes, integers as sum of squares. 

Congruences mod prime powers, 
Hensel's Lemma and its generalization, 
applying it to find solutions of congruences.