yes you can....
so we have
3 mod 15
6 mod 224
gcd(224,15)=1 so we use chinese thm.
first thing we need is integers a and b s.t...
and after we find a and b using euclid's algorithm we compute
6a15+3b224 (mod 15.224=3360)
euclid's algorithm gives a=15 and b=-1
so 6(15)15+3(-1)224=678 mod 3360
and adding and subtracting 3360 would give you many solutions.
you can also use this method to solve many problems which your method would be increasingly difficult to use for example....
Find a number which is 3 mod 7, 2 mod 5 and 1 mod 2.
7,5 and 2 are coprime so use chinese thm...
first concentrate on 3 mod 7 and 2 mod 5
find a,b so that a7+b5=1
a=-2 and b=3 using euclid
compute 2a7+3b5 (mod 7.5=35)=2(-2)7+3(3)5=17 mod 35
now do the same with 17 mod 35 and 1 mod 2
i need to compute 1a35+17b2 where a35+b2=1
a=1 and b=-17
so 1a35+17b2=17 mod 70=87mod70
Note: the reason why we need the gcd to be 1 is because the chinese remainder theorem is based on this.
but you have a nice little argument there, nice one.