For the question,

Find the smallest x which solves the system of congruence equations:

x =+ 3 mod 15

x =+ 6 mod 224, where =+ stands for congruence modulo operation.

I gave an argument oriented solution as follows.

Any multiple of 224 is even.

So, 6+ any multiple of 224 is even.

=> x is even.

Also, any multiple of 15 ends with 5 or 0.

3+any multiple of 15 ends with 8 or 3.

Since X is even, the number should end with 8.

Since X = A number ending with 8,

Also, X = A multiple of 224 + 6

This means the multiple of 224 ends with 2.

Checking the multiples of 224, we get 224, 448, 672.

The smallest one is 672 ==> X is 678.

This solves both the equations as 678 = 224*3 +6

= 45*15 +3

My friend argued about a general solution for the same problem using Chinese Remainder Theorem. I browsed results for the same in various sites.

I find only a general solution found in most cases.

Can anyone describe how to get the particular solution for this problem through Chinese Remainder Theorem?

Also, please mention if there are any mistakes in my argument-oriented solution.