Question

Tanner and Keith are shelving books at a public library. Tanner shelves 4 books at a time, whereas Keith shelves 8 at a time. If they end up shelving the same number of books, what is the smallest number of books each could have shelved?

Answer 1

Let's denote the number of books Tanner shelved as \( T \) and the number of books Keith shelved as \( K \). According to the problem, Tanner shelves 4 books at a time, and Keith shelves 8 books at a time. Thus, we can express the number of books shelved by each as follows:

\[
T = 4m \quad \text{for some integer } m
\]
\[
K = 8n \quad \text{for some integer } n
\]

We need to find the smallest number of books \( T \) and \( K \) such that \( T = K \). This leads to the equation:

\[
4m = 8n
\]

We can simplify this equation by dividing both sides by 4:

\[
m = 2n
\]

From this equation, we can see that \( m \) must be an even integer (since it equals 2 times some integer \( n \)). Therefore, we can express \( m \) in terms of \( n \):

\[
m = 2n
\]

Substituting \( m \) back into the equation for \( T \):

\[
T = 4m = 4(2n) = 8n
\]
\[
K = 8n
\]

From this relationship, we conclude that both \( T \) and \( K \) are equal to \( 8n \). To find the smallest number of books shelved by both Tanner and Keith, we take \( n = 1 \):

\[
T = K = 8 \cdot 1 = 8
\]

Thus, the smallest number of books that both Tanner and Keith could have shelved, such that they shelve the same number of books, is

\[
\boxed{8}.
\]