### Instruction: Let $r$ be the remainder when $1342$ is divided by $13$.

Determine the smallest positive integer that has these two properties:

$\bullet~$ It is a multiple of $1342$.

$\bullet~$ Its remainder upon being divided by $13$ is smaller than $r$.### Response: To find the remainder when $1342$ is divided by $13$, I can use long division or the fact that $1342 = 13 \cdot 103 + 3$.
So $r = 3$.
I need to find a multiple of $1342$ that has a smaller remainder than $3$ when divided by $13$.
One way to do this is to look at the multiples of $1342$ and see what their remainders are.
The first multiple of $1342$ is $1342$ itself, which has a remainder of $3$.
The second multiple of $1342$ is $1342 \cdot 2 = 2684$, which has a remainder of $6$.
The third multiple of $1342$ is $1342 \cdot 3 = 4026$, which has a remainder of $9$.
The fourth multiple of $1342$ is $1342 \cdot 4 = 5368$, which has a remainder of $12$.
The fifth multiple of $1342$ is $1342 \cdot 5 = 6710$, which has a remainder of $2$.
So the smallest positive integer that has the desired properties is $6710$.