How many three-digit positive integers $N$ satisfy the following properties?

The number $N$ is divisible by $7$.
The number formed by reversing the digits of $N$ is divisible by $5$.
$\textbf{(A) } 13 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 16 \qquad \textbf{(E) } 17$
Multiples of $5$ will always end in $0$ or $5$, and since the numbers have to be a three-digit numbers (otherwise it would be a two-digit number), it cannot start with 0, narrowing our choices to 3-digit numbers starting with $5$. Since the numbers must be divisible by 7, all possibilities have to be in the range from $7 \cdot 72$ to $7 \cdot 85$ inclusive. 
$85 - 72 + 1 = 14$. $\boxed{\textbf{(B) } 14}$.
~walmartbrian ~Shontai ~andliu766 ~andyluo ~ESAOPS