Congruences With Composite Moduli

To solve a congruence with composite modulus, write the modulus as a product of prime powers, then solve the congruence simultaneously for modulus of each prime power.,br /> Example: Solve  
\[2x^2+5x \equiv 3 \; (mod \; 72)\]
.
\[72 =8 \times 9=2^3 \times 3^2\]
  so the congruence becomes
\[2x^2+5x \equiv 3 \; (mod \; 8)\]
  (1)
\[2x^2+5x \equiv 3 \; (mod \; 9)\]
  (2)
Solving (1) by exhaustion, trying  
\[x \equiv 0, \; 1, \; 2,..., \; 7 \; (mod \; 8)\]
  gives  
\[x \equiv 5 \; (mod \; 8\]
  as the only solution.
Then  
\[x \equiv 5, \; 13, \; 21, \; 29, \; 37, \; 45, \; 53, \; 61, \; 69 \; (mod \; 72)\]
  are possible solutions to the question.
Solving (2) by exhaustion, trying  
\[x \equiv 0, \; 1, \; 2,..., \; 8 \; (mod \; 9)\]
  gives  
\[x \equiv 5, \; 6 \; (mod \; 9\]
  as the only solutions.
Then  
\[x \equiv 5, \; 6, \; 14, \; 15, \; 23, \; 24, \; 32, \; 33, \; 41, \; 42, \; 50, \; 51 \; 59, \; 60 \; 68, \; 69 \; (mod \; 72)\]
  are possible solutions to the question.
The numbers in both lists of possible solutions 5, 69.