You only have sticks that are 6 and $7 \mathrm{~cm}$ long. What is the minimum number of these sticks needed to cover a 2-meter line segment?
(a) 29
(b) 30
(c) 31
(d) 32
(e) 33
The correct option is (a).

The number of sticks is minimal when the number of $7 \mathrm{~cm}$ sticks used is as large as possible. The segment measures $200 \mathrm{~cm}$. Dividing 200 by 7, we get $200 = 28 \times 7 + 4$. Therefore, if we tried to use only $7 \mathrm{~cm}$ sticks, we would need 29 sticks, but there would still be $3 \mathrm{~cm}$ left over. To avoid having these $3 \mathrm{~cm}$ left over, we can simply replace 3 of the 29 $7 \mathrm{~cm}$ sticks with $6 \mathrm{~cm}$ sticks. We have $26 \times 7 + 3 \times 6 = 200$. Thus, the minimum number of sticks is $26 + 3 = 29$. We should use 26 sticks of $7 \mathrm{~cm}$ and 3 sticks of $6 \mathrm{~cm}$.

Note: Observe that the solution is equivalent to finding integers $x$ and $y$ such that $200 = \underbrace{7 y}_{\text{multiple of } 7} + \underbrace{6 x}_{\text{multiple of } 6}$ and $y$ is as large as possible, where $y$ denotes the number of $7 \mathrm{~cm}$ sticks and $x$ the number of $6 \mathrm{~cm}$ sticks.