A book is said to have $n$ leaves if it is composed of $n$ pieces of paper. On the other hand, the number of pages is twice the number of leaves because each side of a piece of paper is defined as a page.

If the number of pages in a book is $3$ more than a multiple of $7$, and the number of leaves is greater than $100$, then what is the smallest possible number of leaves?
To find the smallest possible number of leaves, I want to find the smallest possible number of pages that satisfies the given conditions. Since the number of pages is $3$ more than a multiple of $7$, I can write it as $7k + 3$ for some non-negative integer $k$. The number of leaves is half the number of pages, so it is $\frac{7k + 3}{2}$. This must be an integer greater than $100$, so I can write the inequality as

\[\frac{7k + 3}{2} > 100\] To solve for $k$, I can multiply both sides by $2$ and subtract $3$, getting

\[7k > 197\] Dividing both sides by $7$, I get

\[k > \frac{197}{7} \approx 28.14\] Since $k$ must be an integer, the smallest possible value for $k$ is $29$. This means the smallest possible number of pages is $7(29) + 3 = 206$. The smallest possible number of leaves is half of that, which is $103$. 