## Congruences With Composite Moduli

\[2x^2+5x \equiv 3 \; (mod \; 72)\]

.\[72 =8 \times 9=2^3 \times 3^2\]

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.