A baker went to the market to buy eggs to make 43 cakes, all with the same recipe, which requires fewer than nine eggs. The seller notices that if he tries to wrap the eggs the baker bought in groups of two, or three, four, five, or six eggs, there is always one egg left over. How many eggs does she use in each cake? What is the minimum number of eggs the baker will use to make the 43 cakes?
Since the 43 cakes have the same recipe, the number of eggs the baker needs is a multiple of 43. On the other hand, this number is also a multiple of 2, 3, 4, 5, and 6, plus 1. The LCM of $2,3,4,5$ and 6 is 60, but $60+1=61$ is not a multiple of 43. We need to find a number with these two properties:

- it is a multiple of 43;
- when increased by 1, it is a multiple of $2,3,4,5$ and 6.

Remember also that since the recipe uses fewer than nine eggs, the number we are looking for is less than $43 \times 9=387$. We have:

$$
\begin{array}{ll}
60 \times 2+1=121 & \text { is not a multiple of } 43 \\
60 \times 3+1=181 & \text { is not a multiple of } 43 \\
60 \times 4+1=241 & \text { is not a multiple of } 43 \\
60 \times 5+1=301 & \text { is a multiple of } 43 \\
60 \times 6+1=361 & \text { is not a multiple of } 43
\end{array}
$$

We can stop here because the next numbers will be greater than 387. Therefore, the baker bought exactly 301 eggs.