elementary number theory - Computing Linear Congruence -Uncertainty Of Answer

given:
$$17x\equiv 3\pmod{2\cdot3\cdot5\cdot7}$$

after extended euclidean algorithm of $(17,210)$ I got:

$$1=3\cdot210+17\cdot(-37)$$ now multplying both RHS and LHS by $3$ I get :

$$210\cdot9+17\cdot(-111)=3$$

then, doing (mod $210$) for both sides I get:

$$17\cdot(-111)\equiv 3\pmod{210}$$

so the answer supposed to be $-111$ but the answer is $99$ for $X.$

My question: is it valid to add $210$ just to $-111$ without the all exp. which is $17(-111)$?