We have a number $X$ between $4899$ and $7330$. When divided by $11$ the number $X$ leaves behind $3$, and when it is divided by $13$ it leaves behind $2$, and finally when divided by $17$ the remainder would be $7$. What is $X$?
How do I go about this using the Chinese remainder theorem? I am able to obtain $1367$ but it not between $4899$ and $7330$.
Chinese Remainder Theorem tells you that all solutions have the form $1367 + k \cdot 11 \cdot 13 \cdot 17$. Now, $11 \cdot 13 \cdot 17=2431$, so that $$1367+2431=3798$$ $$1367+ 2 \cdot 2431 = 6229$$ $$1367+ 3 \cdot 2431 = 8660$$ are other solutions (you can go on and write all other solutions). The unique belonging to your range is $\mathbf{6229}$, and this is what you were looking for.