Find the greatest common divisor for the modulus and the coefficient of the variable, for each congruence. If the remainder of the congruence is divisible by that greatest common divisor, then the congruence has a solution. Furthermore, for ax≡b (mod n), the greatest common divisor for n and a is the number of solutions for x when x lies between 0 and n, inclusive. For instance, in the congruence 4x≡2 (mod 6), the greatest common divisor of 6 and 4 is 2, and 2 -- the remainder -- is divisible by 2 -- the greatest common divisor, so this congruence has 2 solutions. If any of the congruences have no solutions, then there is no solution to the system.
Solve the first congruence by using the formula x=kn+b, where n is the modulus and and b is the remainder. If the congruences are x≡3 (mod 5) and x≡5 (mod 8), then x=5k+3 for the first congruence.
Substitute the value of x in terms of k into the next equation. In the example above, this would mean that 5k+3≡5 (mod 8). Subtracting 3 from both sides yields 5k≡2 (mod 8). Solve for k by adding 8, the modulus, to 2, the remainder, until you reach a number divisible by 5. In this case, 2+8=10, and 10 is divisible by 5, so 5k≡10 (mod 8). Divide through by 5 to get k≡2 (mod 8), or k=8m+2. If there are more than two congruences, then repeat this step for each additional congruence, using a new variable for each congruence.
Substitute the value of k from the second congruence into the value for x in the first congruence to find a value for x that will work for both congruences. In the example above, because k=8m+2, then x=5(8m+2)+3=40m+13, or x≡13 (mod 40). This means the values for x that will work for both x≡3 (mod 5) and x≡5 (mod 8) are 13, 53, 93, etc.