How do you apply the Chinese Remainder Theorem to this problem?

Question

The simultaneous congruences are

2x congruent to 1(mod 3)
5x congruent to 6(mod 7)
3x congruent to 4(mod 8)

Please give any workings

Answer 1

Lets first solve each equation separately::
2x = 1 (mod 3)
x = 2^-1 * 1 (mod 3)
-----------------------> x = 2 (mod 3)


5x = 6 (mod 7)
x = 5^-1 * 6 = 3 * 6 (mod 7)
-----------------------> x = 4 (mod 7)


3x = 4 (mod 8)
x = 3^-1 * 4 = 3 *4 (mod 8)
-----------------------> x = 4 (mod 8)

Now we can rewrite the original system of equations
in Chinese Remainder Theorem canonical form:

x = 2 (mod 3)
x = 4 (mod 7)
x = 4 (mod 8)

Hope it helps.