why there is not such number x in N:
x=2(mod6)
x=0(mod15)
x=4(mod7)
If $\displaystyle x=2\!\!\!\pmod 6$ then $\displaystyle x$ is even, and if also $\displaystyle x=0\!\!\!\pmod {15}$ then it must be an even multiple of 15. But even multiples of 15 are multiples of 6 so...
The CRT doesn't apply here since $\displaystyle gcd(6,15)\neq 1$
Tonio